The Derivative and Differentiation

This is meant to be a theoretical treatment of differentiation and all of its related concepts.  These would have been covered in a standard Calculus course, but here we will endeavor to include proofs of the main results.

I.       Rates of Change and Tangent Lines

When a particle or a person is moving, or in motion, we can measure the ratio of how far he/she/it has traveled to how long it takes to travel that distance.  This ratio is called the average speed.  The units of measure are units of length divided by units of time – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@  miles per hour, feet per second, furlongs per fortnight, or whatever is important to the problem at hand.

Below are some examples of different average speeds.

Notation: m/s = meters per second; km/h = kilometers per hour; mph = miles per hour; fps = feet per second

  1. Speed of a common snail: 0.001 m/s = 0.0036 km/h = 0.0023 mph = 0.0188 inches per second.
  2. A brisk walk = 1.667 m/s = 6 km/h = 3.75 mph.
  3. Olympic sprinters (average speed over 100 meters) = 10.23 m/s = 36.85 km/h =23.11 mph.
  4. Speed limit on an interstate highway = 28.794 m/s = 103.66 km/h = 65 mph = 44.32 fps.
  5. Top cruising speed of a Boeing 747-8 = 290.947 m/s = 1047.41 km/h = 650.83 mph : (officially Mach 0.85)
  6. Official air speed record = 980.278 m/s = 3,529 km/h = 2,188 mph.
  7. Space shuttle on re-entry = 7,777.778 m/s = 28,000 km/h = 17,500 mph.
  8. the speed of sound in dry air at 70°F (Mach 1): 344 m/s = 1238 km/h = 769 mph = 1128 fps
  9. The speed of sound in water at 25°C: 1,497 m/s = 5,389.2 km/h = 3,379.31 mph
  10. The elevators in the Sears Tower, Chicago (1451 ft): 1600 ft/min
  11. Highest surface wind speed recorded on earth: 231 mph at Mt. Washington Observatory, NH
  12. F-22A Raptor supersonic fighter: Mach 2.5+, in excess of 1600 mph
  13. Dodge Viper GTS Coupe: 177 mph
  14. 81 mph – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@  fastest human powered bicycle speed on flat ground
  15. Top speed of the cheetah – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@  70 mph
  16. Top speed of the Thompson’s gazelle (the cheetah’s favorite food) – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@  50 mph
  17. Top speed of the Peregrine falcon – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@  200 mph
  18. Top speed of the Sailfish (genus Istiphorus) – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@  68.5 mph

There are clearly many different average speeds to consider.  Let’s say that you are climbing at Hanging Rock State Park near Danbury, NC.  A rock breaks loose from one of the cliffs.  What would be the average speed during the first 2 seconds of the fall?

From physics we know that a dense solid object dropped from rest that falls freely near the surface of the earth will fall according to the equation:

y=16 t 2 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2da9iaaigdacaaI2aGaamiDamaaCaaaleqabaaeaaaaaaaaa8qacaaIYaaaaaaa@3B74@  feet

in the first t seconds.  Thus the average speed of the rock over the first two seconds will be the distance it falls, Δy MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaamyEaaaa@3858@ , divided by the time differential, Δt MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaamiDaaaa@3853@ , so for the average speed we get

Δy Δt = 16 (2) 2 −16 (0) 2 2−0 =32 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHuoarcaWG5baabaGaeyiLdqKaamiDaaaacqGH9aqpdaWcaaqaaiaaigdacaaI2aGaaiikaiaaikdacaGGPaWaaWbaaSqabeaaqaaaaaaaaaWdbiaaikdaaaGccqGHsislcaaIXaGaaGOnaiaacIcacaaIWaGaaiyka8aadaahaaWcbeqaa8qacaaIYaaaaaGcpaqaaiaaikdacqGHsislcaaIWaaaaiabg2da9iaaiodacaaIYaaaaa@4AFF@  ft/sec

This tells us the average speed.  However, we know that the rock is picking up speed as it falls, so this does not tell us what the speed is at any one moment in time.  For that we want to find the instantaneous speed.  That is, we need to find the speed over a smaller and smaller time period. So, at the instant when t=2, the speed of the rock will be about equal to the average speed over a very short time period, or between t=2 and t=2+h MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabg2da9iaaikdacqGHRaWkcaWGObaaaa@3A7D@  for a very small number h.

Δy Δt = 16 (2+h) 2 −16 (2) 2 (2+h)−2 = 16 (2+h) 2 −16 (2) 2 h MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHuoarcaWG5baabaGaeyiLdqKaamiDaaaacqGH9aqpdaWcaaqaaiaaigdacaaI2aGaaiikaiaaikdacqGHRaWkcaWGObGaaiykamaaCaaaleqabaaeaaaaaaaaa8qacaaIYaaaaOGaeyOeI0IaaGymaiaaiAdacaGGOaGaaGOmaiaacMcapaWaaWbaaSqabeaapeGaaGOmaaaaaOWdaeaacaGGOaGaaGOmaiabgUcaRiaadIgacaGGPaGaeyOeI0IaaGOmaaaacqGH9aqpdaWcaaqaaiaaigdacaaI2aGaaiikaiaaikdacqGHRaWkcaWGObGaaiykamaaCaaaleqabaWdbiaaikdaaaGccqGHsislcaaIXaGaaGOnaiaacIcacaaIYaGaaiyka8aadaahaaWcbeqaa8qacaaIYaaaaaGcpaqaaiaadIgaaaaaaa@5B7E@  ft/sec

We cannot use this to find the speed at t=2 because that would mean that we would have to take h=0, and this would give us 0/0, which is undefined.  Thus, we want to know the value of this quotient when h is close to 0 and getting closer to 0, or

lim h→0 16 (2+h) 2 −16 (2) 2 (2+h)−2 = lim h→0 16 (2+h) 2 −16 (2) 2 h = lim h→0 (64+16h) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7179@  ft/sec

I.1: Average Rate of Change

For a given function f(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaaaa@3934@  we can compute the average rate of change of the function over the interval [a,b] by the difference quotient:

f(b)−f(a) b−a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGMbGaaiikaiaadkgacaGGPaGaeyOeI0IaamOzaiaacIcacaWGHbGaaiykaaqaaiaadkgacqGHsislcaWGHbaaaaaa@3FFF@

 

Year

National Defense Spending
 ($ billion)

 

1990

299.3

 

1995

272.1

 

1999

274.9

 

2000

294.5

 

2001

305.5

 

2002

348.6

 

2003

404.9

 

2004

455.9

 

2005

495.3

 

2006

535.9

 

In the table and graph above we see the National Defense Spending in a number of different years since 1990.  We can find the average rate of change in the defense spending between 1999 and 2004.  If we let P=(1999,274.39) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiabg2da9iaacIcacaaIXaGaaGyoaiaaiMdacaaI5aGaaiilaiaaikdacaaI3aGaaGinaiaac6cacaaIZaGaaGyoaiaacMcaaaa@4148@  and Q=(2004,455.9) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuaiabg2da9iaacIcacaaIYaGaaGimaiaaicdacaaI0aGaaiilaiaaisdacaaI1aGaaGynaiaac6cacaaI5aGaaiykaaaa@4077@ , then this rate of change is given by the quotient

Average rate of change: Δy Δx = 455.9−274.9 2004−1999 = 181 5 =36.2 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaqaaaaaaaaaWdbiabfs5aejaadMhaa8aabaWdbiabfs5aejaadIhaaaWdaiabg2da98qadaWcaaWdaeaapeGaaGinaiaaiwdacaaI1aGaaiOlaiaaiMdacqGHsislcaaIYaGaaG4naiaaisdacaGGUaGaaGyoaaWdaeaapeGaaGOmaiaaicdacaaIWaGaaGinaiabgkHiTiaaigdacaaI5aGaaGyoaiaaiMdaaaGaeyypa0ZaaSaaa8aabaWdbiaaigdacaaI4aGaaGymaaWdaeaapeGaaGynaaaacqGH9aqpcaaIZaGaaGOnaiaac6cacaaIYaaaaa@53E5@  billion dollars per year

Note that this is the slope of the secant line joining P to Q.  In fact, we can always think of the average rate of change as being the slope of a secant line.

Q

Slope of PQ=Δy/Δx

(2003,404.9)

(404.9−274.9)/(2003−1999)=32.5 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaKaaiikaiaaisdacaaIWaGaaGinaiaac6cacaaI5aGaeyOeI0IaaGOmaiaaiEdacaaI0aGaaiOlaiaaiMdacaGGPaGaai4laiaacIcacaaIYaGaaGimaiaaicdacaaIZaGaeyOeI0IaaGymaiaaiMdacaaI5aGaaGyoaiaacMcacqGH9aqpcaaIZaGaaGOmaiaac6cacaaI1aaaaa@4CB7@

(2002,348.6)

(348.6−274.9)/(2002−1999)=24.567 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaKaaiikaiaaiodacaaI0aGaaGioaiaac6cacaaI2aGaeyOeI0IaaGOmaiaaiEdacaaI0aGaaiOlaiaaiMdacaGGPaGaai4laiaacIcacaaIYaGaaGimaiaaicdacaaIYaGaeyOeI0IaaGymaiaaiMdacaaI5aGaaGyoaiaacMcacqGH9aqpcaaIYaGaaGinaiaac6cacaaI1aGaaGOnaiaaiEdaaaa@4E3C@

(2001,305.5)

(305.5−274.9)/(2001−1999)=15.3 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaKaaiikaiaaiodacaaIWaGaaGynaiaac6cacaaI1aGaeyOeI0IaaGOmaiaaiEdacaaI0aGaaiOlaiaaiMdacaGGPaGaai4laiaacIcacaaIYaGaaGimaiaaicdacaaIXaGaeyOeI0IaaGymaiaaiMdacaaI5aGaaGyoaiaacMcacqGH9aqpcaaIXaGaaGynaiaac6cacaaIZaaaaa@4CB0@

(2000,294.5)

(294.5−274.9)/(2000−1999)=19.6 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaKaaiikaiaaikdacaaI5aGaaGinaiaac6cacaaI1aGaeyOeI0IaaGOmaiaaiEdacaaI0aGaaiOlaiaaiMdacaGGPaGaai4laiaacIcacaaIYaGaaGimaiaaicdacaaIWaGaeyOeI0IaaGymaiaaiMdacaaI5aGaaGyoaiaacMcacqGH9aqpcaaIXaGaaGyoaiaac6cacaaI2aaaaa@4CBD@

We might want to know the average rates of change between 1999 and 2003, or between 1999 and 2002, etc.  These are the slopes of the appropriate secant lines.  We can compute these slopes, or average rates of change.

As we let Q get closer and closer to the P, we see that the secant lines get closer and closer to the line that is tangent to the curve at the point P.  If we were to sketch in a line that we think might approximate the tangent line, we might find that it passes through (1994,250) and (2006,310) and the slope of that line is 5 billion dollars per year.  This, of course is strictly an approximation.  It does, however, give us the idea behind what we might want to define for the tangent line and how we might be able to compute its slope.  We clearly have the point (a,f(a)) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggacaGGSaGaamOzaiaacIcacaWGHbGaaiykaiaacMcaaaa@3C0C@  through which the tangent line would pass.  The only other information we need in order to determine uniquely that tangent line would be its slope.  If the slope were mT, then the equation of the tangent line would be y−f(a)= m T (x−a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabgkHiTiaadAgacaGGOaGaamyyaiaacMcacqGH9aqpcaWGTbWaaSbaaSqaaabaaaaaaaaapeGaamivaaWdaeqaaOGaaiikaiaadIhacqGHsislcaWGHbGaaiykaaaa@4267@  or y=f(a)+ m T (x−a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2da9iaadAgacaGGOaGaamyyaiaacMcacqGHRaWkcaWGTbWaaSbaaSqaaabaaaaaaaaapeGaamivaaWdaeqaaOGaaiikaiaadIhacqGHsislcaWGHbGaaiykaaaa@425C@ .

I.2: Tangent Line to a Curve

We use the terminology from geometry.  Given a line and a circle, there are exactly three ways they might intersect:

1.       They can intersect in no points

2.      They can intersect in one point, in which case we say that the line is tangent to the circle.

3.      They can intersect in two points, in which case we call the line a secant line.

Now, for the most part, calculus is the study of local phenomena.  What we mean by that is that we study what happens over small intervals around certain points.  We do, occasionally, study global phenomena, but usually we are interested in what is going on in a small neighborhood.  In our definition of secants and tangents we really need to accede to that point of view.  We are going to choose a point (a,f(a)) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggacaGGSaGaamOzaiaacIcacaWGHbGaaiykaiaacMcaaaa@3C0C@  on the curve.  In terms of nearby points, or in a small neighborhood, a secant line to the curve is a line that passes through (a,f(a)) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggacaGGSaGaamOzaiaacIcacaWGHbGaaiykaiaacMcaaaa@3C0C@  and (b,f(b)) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadkgacaGGSaGaamOzaiaacIcacaWGIbGaaiykaiaacMcaaaa@3C0E@ , where b is usually relatively close to a.  A tangent line to f(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaaaa@3934@  at x=a is the limit of the secant lines as b approaches a.  Visually this is a line the touches the graph of y=f(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2da9iaadAgacaGGOaGaamiEaiaacMcaaaa@3B38@  only once in a small neighborhood about a, but does so in a manner that tends to behave like f(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaaaa@3934@ .

How can we see this?  How can we make this more visual?  Think about the graph of your favorite function.  Let’s say that we want to look at the graph of f(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaaaa@3934@  on [-10,10]:

Graph of y = 2*sin(Pi*x)*e^(-x^2/125) from -10 to 10

Figure 1: f(x)=2sin(πx) e − x 2 /125 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iaaikdaqaaaaaaaaaWdbiGacohacaGGPbGaaiOBaiaacIcacqaHapaCcaWG4bGaaiykaiaadwgapaWaaWbaaSqabeaapeGaeyOeI0IaamiEa8aadaahaaadbeqaa8qacaaIYaaaaSGaai4laiaaigdacaaIYaGaaGynaaaaaaa@491E@

Now, this graph looks pretty “wiggly”.  It oscillates quickly and with different heights.  We are interested in what is happening around x=0.  So let’s zoom in on the graph a few times – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@  each time by a factor of 10.  Thus, our first zoom will be to the interval [-1,1] and then to [-0.1,0.1], and so on.

  Graph of y = 2*sin(Pi*x)*e^(-x^2/125) from -1 to 1   Graph of y = 2*sin(Pi*x)*e^(-x^2/125) from -0.1 to 0.1   Graph of y = 2*sin(Pi*x)*e^(-x^2/125) from -0.01 to 0.01   Graph of y = 2*sin(Pi*x)*e^(-x^2/125) from -0.001 to 0.001
            f(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaaaa@3934@  on [-1,1]                     f(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaaaa@3934@ on [-0.1,0.1]              f(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaaaa@3934@ on [-0.01,0.01]         f(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaaaa@3934@ on [-0.001,0.001]

As we zoom in we see what happens to the graph – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@  it “flattens out”.  It begins to look like a line.  By the time we are down to the interval [ – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@ 0.1,0.1] it already looks like a line, and this is certainly true by the time we are looking on the interval [ – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@ 0.001,0.001]!  The graph looks linear, and we describe this by saying that the function is locally linear if this happens.  This line that the function looks like should be the tangent line, if there is one.

Let’s see if we can’t compute one of these.  Let’s look at the function f(x)= 1 2 x 2 −3x+5 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaiabg2da9maalmaaleaacaaIXaaabaGaaGOmaaaakiaadIhadaahaaWcbeqaaabaaaaaaaaapeGaaGOmaaaakiabgkHiTiaaiodacaWG4bGaey4kaSIaaGynaaaa@4231@  at the point (2,f(2))=(2,1) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaikdacaGGSaGaamOzaiaacIcacaaIYaGaaiykaiaacMcacqGH9aqpcaGGOaGaaGOmaiaacYcacaaIXaGaaiykaaaa@403E@ .  The graph of this function on the interval [0,5] looks like the following.

Let’s compute the slope of the secant line that runs from (2,1) to the point (2+h,f(2+h)) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaikdacqGHRaWkcaWGObGaaiilaiaadAgacaGGOaGaaGOmaiabgUcaRiaadIgacaGGPaGaaiykaaaa@3F56@ . 

Secant slope= f(2+h)−f(2) h MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4uaiaabwgacaqGJbGaaeyyaiaab6gacaqG0bGaaeiiaiaabohacaqGSbGaae4BaiaabchacaqGLbGaeyypa0ZaaSaaaeaacaWGMbGaaiikaiaaikdacqGHRaWkcaWGObGaaiykaiabgkHiTiaadAgacaGGOaGaaGOmaiaacMcaaeaacaWGObaaaaaa@4B77@

= ( 1 2 (2+h) 2 −3(2+h)+5)−1 h MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaSaaaeaacaGGOaWaaSqaaSqaaiaaigdaaeaacaaIYaaaaOGaaiikaiaaikdacqGHRaWkcaWGObGaaiykamaaCaaaleqabaaeaaaaaaaaa8qacaaIYaaaaOWdaiabgkHiTiaaiodacaGGOaGaaGOmaiabgUcaRiaadIgacaGGPaGaey4kaSIaaGynaiaacMcacqGHsislcaaIXaaabaGaamiAaaaaaaa@48C9@

= 1 2 h 2 −h h = 1 2 h−1 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaSaaaeaadaWcbaWcbaGaaGymaaqaaiaaikdaaaGccaWGObWaaWbaaSqabeaaqaaaaaaaaaWdbiaaikdaaaGcpaGaeyOeI0IaamiAaaqaaiaadIgaaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaacaWGObGaeyOeI0IaaGymaaaa@429E@

The limit of the secant slope as Q approaches P along the curve is the slope of the tangent line.  In this case we get

lim Q→P (secant slope)= lim h→0 ( 1 2 h−1 )=−1 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaqaaaaaaaaaWdbiGacYgacaGGPbGaaiyBaaWcpaqaa8qacaWGrbGaeyOKH4QaamiuaaWdaeqaaOGaaiikaiaabohacaqGLbGaae4yaiaabggacaqGUbGaaeiDaiaabccacaqGZbGaaeiBaiaab+gacaqGWbGaaeyzaiaacMcacqGH9aqpdaWfqaqaa8qaciGGSbGaaiyAaiaac2gaaSWdaeaapeGaamiAaiabgkziUkaaicdaa8aabeaakmaabmaabaWdbmaalaaapaqaa8qacaaIXaaapaqaa8qacaaIYaaaaiaadIgacqGHsislcaaIXaaapaGaayjkaiaawMcaa8qacqGH9aqpcqGHsislcaaIXaaaaa@59D5@ .

Thus the slope of the parabola at (2,1) is – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@ 1.  The tangent line to the curve at P is the line through (2,1) with slope m= – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@ 1.

y−1=−1(x−2) y=−x+3 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacaWG5bGaeyOeI0IaaGymaiabg2da9iabgkHiTiaaigdacaGGOaGaamiEaiabgkHiTiaaikdacaGGPaaabaGaamyEaiabg2da9iabgkHiTiaadIhacqGHRaWkcaaIZaaaaaa@44DD@

The slope of a curve y=f(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2da9iaadAgacaGGOaGaamiEaiaacMcaaaa@3B38@  at the point (a,f(a)) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggacaGGSaGaamOzaiaacIcacaWGHbGaaiykaiaacMcaaaa@3C0C@  is the number

lim h→0 f(a+h)−f(a) h = lim x→a f(x)−f(a) x−a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaqaaaaaaaaaWdbiGacYgacaGGPbGaaiyBaaWcpaqaa8qacaWGObGaeyOKH4QaaGimaaWdaeqaaOWdbmaalaaapaqaa8qacaWGMbGaaiikaiaadggacqGHRaWkcaWGObGaaiykaiabgkHiTiaadAgacaGGOaGaamyyaiaacMcaa8aabaWdbiaadIgaaaGaeyypa0ZdamaaxababaWdbiGacYgacaGGPbGaaiyBaaWcpaqaa8qacaWG4bGaeyOKH4QaamyyaaWdaeqaaOWdbmaalaaapaqaa8qacaWGMbGaaiikaiaadIhacaGGPaGaeyOeI0IaamOzaiaacIcacaWGHbGaaiykaaWdaeaapeGaamiEaiabgkHiTiaadggaaaaaaa@59FF@

if the limit exists.  The tangent line to the curve at the point is the line through the point (a,f(a)) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggacaGGSaGaamOzaiaacIcacaWGHbGaaiykaiaacMcaaaa@3C0C@  with the above slope.

 

II Definition of the Derivative

In the last section we defined the slope of a curve y=f(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2da9iaadAgacaGGOaGaamiEaiaacMcaaaa@3B38@  at the point x=a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iaadggaaaa@38DC@  to be the limit of the difference quotient:

m= lim h→0 f(a+h)−f(a) h MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2da9maaxababaaeaaaaaaaaa8qaciGGSbGaaiyAaiaac2gaaSWdaeaapeGaamiAaiabgkziUkaaicdaa8aabeaak8qadaWcaaWdaeaapeGaamOzaiaacIcacaWGHbGaey4kaSIaamiAaiaacMcacqGHsislcaWGMbGaaiikaiaadggacaGGPaaapaqaa8qacaWGObaaaaaa@493B@

When this limit exists, that number is called the derivative of f at a.  We can study this phenomenon at each point in the domain of f at which the limit exists.

Definition: The derivative of the function f is the function f ′ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaaaaa@36EA@  whose value at each point x in the domain of f is

f ′ (x)= lim h→0 f(x+h)−f(x) h MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaaiikaiaadIhacaGGPaGaeyypa0ZaaCbeaeaaqaaaaaaaaaWdbiGacYgacaGGPbGaaiyBaaWcpaqaa8qacaWGObGaeyOKH4QaaGimaaWdaeqaaOWdbmaalaaapaqaa8qacaWGMbGaaiikaiaadIhacqGHRaWkcaWGObGaaiykaiabgkHiTiaadAgacaGGOaGaamiEaiaacMcaa8aabaWdbiaadIgaaaaaaa@4BC4@

provided that the limit exists.

Note that the domain of f ′ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaaaaa@36EA@  is the set of points in the domain of f at which the limit exists.  This may be strictly smaller than the domain of f, but can be no larger than the domain of f.  If f ′ (x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaaiikaiaadIhacaGGPaaaaa@3940@  exists, then we say that f is differentiable at x. A function that is differentiable at each point in its domain is a differentiable function.

We denote the derivative of f in a couple of different ways.  Each notation has its strong points and its drawbacks.  The most common notation for the derivative of f at x is f ′ (x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaaiikaiaadIhacaGGPaaaaa@3940@ , and is often called “f prime of x”.  This is very close to the notation used by Newton when he worked with “the calculus”.  His notation is still used by some physicists today.  His notation for the derivative of f at x is f ˙ (x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaacaGaaiikaiaadIhacaGGPaaaaa@393D@   — MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa8hfGaaa@3A98@  that is a small dot over f.  It can be hard to see, so most books will use the “prime notation”.  The notation that Leibniz introduced is used today as well.  His notation for the derivative of f at x is df dx (x)= d  dx f(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qadaWcaaWdaeaapeGaamizaiaadAgaa8aabaWdbiaadsgacaWG4baaaiaacIcacaWG4bGaaiykaiabg2da9maalaaapaqaa8qacaWGKbGaaGjbVdWdaeaapeGaamizaiaadIhaaaGaamOzaiaacIcacaWG4bGaaiykaaaa@4562@ .  This is sometimes read as “df by dx”.  It is not a fraction.  A more modern notation introduced in the twentieth century is the notation Df or D x f MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBaaaleaaqaaaaaaaaaWdbiaadIhaa8aabeaak8qacaWGMbaaaa@3919@ . Thus, we have

f ′ (x)= f ˙ (x)= df dx (x)= d  dx f(x)=Df(x)= D x f(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaaiikaiaadIhacaGGPaGaeyypa0JabmOzayaacaGaaiikaiaadIhacaGGPaGaeyypa0deaaaaaaaaa8qadaWcaaWdaeaapeGaamizaiaadAgaa8aabaWdbiaadsgacaWG4baaaiaacIcacaWG4bGaaiykaiabg2da9maalaaapaqaa8qacaWGKbGaaGjbVdWdaeaapeGaamizaiaadIhaaaGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iaadseacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0Jaamira8aadaWgaaWcbaWdbiaadIhaa8aabeaak8qacaWGMbGaaiikaiaadIhacaGGPaaaaa@5996@

Example: Find the derivative of f(x)= x 2 −4x−4 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iaadIhadaahaaWcbeqaaabaaaaaaaaapeGaaGOmaaaakiabgkHiTiaaisdacaWG4bGaeyOeI0IaaGinaaaa@409D@ .

To do this we need to evaluate the limit of the difference quotient.

f'(x)= lim h→0 f(x+h)−f(x) h = lim h→0 (x+h) 2 −4(x+h)−4−( x 2 −4x−4) h MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6B02@

= lim h→0 2xh+ h 2 −4h h = lim h→0 (2x+h−4)=2x−4 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaCbeaeaaqaaaaaaaaaWdbiGacYgacaGGPbGaaiyBaaWcpaqaa8qacaWGObGaeyOKH4QaaGimaaWdaeqaaOWdbmaalaaapaqaa8qacaaIYaGaamiEaiaadIgacqGHRaWkcaWGObWdamaaCaaaleqabaWdbiaaikdaaaGccqGHsislcaaI0aGaamiAaaWdaeaapeGaamiAaaaacqGH9aqppaWaaCbeaeaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIgacqGHsgIRcaaIWaaapaqabaGccaGGOaWdbiaaikdacaWG4bGaey4kaSIaamiAaiabgkHiTiaaisdacaGGPaGaeyypa0JaaGOmaiaadIhacqGHsislcaaI0aaaaa@5A58@

So we find that f ′ (x)=2x−4 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaaiikaiaadIhacaGGPaGaeyypa0JaaGOmaiaadIhacqGHsislcaaI0aaaaa@3DAA@

Alternative Definition: If we use the alternative definition of the slope of the tangent being the limit of the slopes of the secant lines, then we have that the derivative of f at the point x=a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iaadggaaaa@38DC@  is given by

f ′ (a)= lim x→a f(x)−f(a) x−a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaaiikaiaadggacaGGPaGaeyypa0ZaaCbeaeaaqaaaaaaaaaWdbiGacYgacaGGPbGaaiyBaaWcpaqaa8qacaWG4bGaeyOKH4QaamyyaaWdaeqaaOWdbmaalaaapaqaa8qacaWGMbGaaiikaiaadIhacaGGPaGaeyOeI0IaamOzaiaacIcacaWGHbGaaiykaaWdaeaapeGaamiEaiabgkHiTiaadggaaaaaaa@4BE6@

provided that the limit exists.

 

We find that this is usually the formulation that the derivative takes in most applications.  We will be able to recognize certain problems as derivatives by seeing this type of limit in the process of setting up the problem.

 

Example 1:  We can still evaluate the derivative using this alternative definition.  Let’s see that we would get the same thing as before, so let f(x)= x 2 −4x−4 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iaadIhadaahaaWcbeqaaabaaaaaaaaapeGaaGOmaaaakiabgkHiTiaaisdacaWG4bGaeyOeI0IaaGinaaaa@409D@ .

f'(a)= lim x→a f(x)−f(a) x−a = lim x→a x 2 −4x−4−( a 2 −4a−4) x−a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@66A6@

= lim x→a ( x 2 − a 2 )−4(x−a) x−a = lim x→a (x−a)(x+a)−4(x−a) x−a = lim x→a (x−a)(x+a−4) x−a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7C4E@

Now, as long as we get close to a, but never let x equal a, we have that the term x – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@ a in the numerator and denominator factor out.  Thus,

f ′ (a)= lim x→a (x−a)(x+a−4) x−a = lim x→a (x+a−4)=2a−4 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qaceWGMbGbauaacaGGOaGaamyyaiaacMcacqGH9aqppaWaaCbeaeaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGHbaapaqabaGcpeWaaSaaa8aabaWdbiaacIcacaWG4bGaeyOeI0IaamyyaiaacMcacaGGOaGaamiEaiabgUcaRiaadggacqGHsislcaaI0aGaaiykaaWdaeaapeGaamiEaiabgkHiTiaadggaaaGaeyypa0ZdamaaxababaWdbiGacYgacaGGPbGaaiyBaaWcpaqaa8qacaWG4bGaeyOKH4QaamyyaaWdaeqaaOWdbiaacIcacaWG4bGaey4kaSIaamyyaiabgkHiTiaaisdacaGGPaGaeyypa0JaaGOmaiaadggacqGHsislcaaI0aaaaa@6101@

which is what we got before.

Note that this process requires that we try find a factor of x – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@ a in the numerator to factor out with the denominator, whereas in the first definition we just had to find a common factor of h in each term of the numerator.  Algebraically speaking, the first process may be easier, but the second has it uses.

Example 2:  Find the derivative of f(x)= x MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaiabg2da9abaaaaaaaaapeWaaOaaa8aabaWdbiaadIhaaSqabaaaaa@3B91@  using the alternative definition.

f'(a)= lim x→a f(x)−f(a) x−a = lim x→a x − a x−a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacEcacaGGOaGaamyyaiaacMcacqGH9aqpdaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGHbaapaqabaGcpeWaaSaaa8aabaWdbiaadAgacaGGOaGaamiEaiaacMcacqGHsislcaWGMbGaaiikaiaadggacaGGPaaapaqaa8qacaWG4bGaeyOeI0IaamyyaaaacqGH9aqppaWaaCbeaeaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGHbaapaqabaGcpeWaaSaaa8aabaWdbmaakaaapaqaa8qacaWG4baaleqaaOGaeyOeI0YaaOaaa8aabaWdbiaadggaaSqabaaak8aabaWdbiaadIhacqGHsislcaWGHbaaaaaa@5B41@

= lim x→a x − a x−a = lim x→a x − a x−a · x + a x + a = lim x→a x−a (x−a)( x + a ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6FC2@

f ′ (a)= lim x→a x−a (x−a)( x + a ) = lim x→a 1 x + a = 1 2 a = 1 2 a − 1 2 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qaceWGMbGbauaacaGGOaGaamyyaiaacMcacqGH9aqppaWaaCbeaeaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGHbaapaqabaGcpeWaaSaaa8aabaWdbiaadIhacqGHsislcaWGHbaapaqaa8qacaGGOaGaamiEaiabgkHiTiaadggacaGGPaGaaiikamaakaaapaqaa8qacaWG4baaleqaaOGaey4kaSYaaOaaa8aabaWdbiaadggaaSqabaGccaGGPaaaaiabg2da98aadaWfqaqaa8qaciGGSbGaaiyAaiaac2gaaSWdaeaapeGaamiEaiabgkziUkaadggaa8aabeaak8qadaWcaaWdaeaapeGaaGymaaWdaeaapeWaaOaaa8aabaWdbiaadIhaaSqabaGccqGHRaWkdaGcaaWdaeaapeGaamyyaaWcbeaaaaGccqGH9aqpdaWcaaWdaeaapeGaaGymaaWdaeaapeGaaGOmamaakaaapaqaa8qacaWGHbaaleqaaaaakiabg2da9maalaaapaqaa8qacaaIXaaapaqaa8qacaaIYaaaaiaadggapaWaaWbaaSqabeaapeGaeyOeI0YaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaaaaaaaa@64BC@

Example 3:  Find the derivative of f(x)= x 3 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaiabg2da9abaaaaaaaaapeGaamiEa8aadaahaaWcbeqaa8qacaaIZaaaaaaa@3C60@  using the first definition.

f ′ (x)= lim h→0 f(x+h)−f(x) h = lim h→0 (x+h) 3 − x 3 h MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5D3E@

f ′ (x)= lim h→0 3 x 2 h+3x h 2 + h 3 h = lim h→0 (3 x 2 +3xh+ h 2 )=3 x 2 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@63FE@

III.    Local Linearity

Since we have been using graphing calculators extensively in early courses are familiar with the principle of local linearity. We know from much experience that if we zoom in on just about any section of one of the functions we have studied, the function quickly becomes linear. our study of calculus adds to this base by explicitly expressing this zoomed-in line as the tangent to the curve at x = a, y=f′(a)(x−a)+f(a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadMhacqGH9aqpcaWGMbGaeyOmGiQaaiikaiaadggacaGGPaGaaiikaiaadIhacqGHsislcaWGHbGaaiykaiabgUcaRiaadAgacaGGOaGaamyyaiaacMcaaaa@47CE@ .

Local linearity is a property of differentiable functions that says that if you zoom in on a point on the graph of the function (with equal scaling horizontally and vertically), the graph will eventually look like a straight line with a slope equal to the derivative of the function at the point.

Thus, local linearity is the graphical manifestation of differentiability.

Functions that are locally linear are smooth. Functions are not locally linear at points where they have discontinuities: breaks, jumps, vertical asymptotes, or the like. For example, y = |x| is not locally linear at the origin.

Functions that are differentiable at a point are locally linear there, and vice versa.  Unfortunately, there is no other definition of local linearity. It would be helpful if one could determine whether a function is locally linear at a point and then be able to conclude that is it differentiable. This is not possible. Since zooming in is accomplished using technology, such as a graphing calculator or computer graphing program, one can never be certain that one has zoomed in far enough.  Thus if your function is given to you numerically, you can never be sure that it is differentiable.

On the other hand, if you have an analytic representation of your function (i.e. a formula containing standard functions like powers, trigonometric functions, etc.), then you can check to see if the function is differentiable at the point of interest. Namely, take the derivative at the point of interest and see if it is finite, and if so, it is differentiable and hence locally linear.

IV.    Basic Properties of the Derivative

 

Definition Let f be a real-valued function defined on an open interval containing the point a. We say the f is differentiable at a, or that f has a derivative at a, if the limit

lim x→a f(x)−f(a) x−a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGHbaapaqabaGcpeWaaSaaa8aabaWdbiaadAgacaGGOaGaamiEaiaacMcacqGHsislcaWGMbGaaiikaiaadggacaGGPaaapaqaa8qacaWG4bGaeyOeI0Iaamyyaaaaaaa@4A82@

exists and is finite. We will write f′(a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamyyaiaacMcaaaa@3D95@  for the derivative of f at a:

f′(a)= lim x→a f(x)−f(a) x−a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamyyaiaacMcacqGH9aqppaWaaCbeaeaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGHbaapaqabaGcpeWaaSaaa8aabaWdbiaadAgacaGGOaGaamiEaiaacMcacqGHsislcaWGMbGaaiikaiaadggacaGGPaaapaqaa8qacaWG4bGaeyOeI0Iaamyyaaaaaaa@5051@

whenever this limit exists and is finite.

We will want to speak about f ′ as function in its own right. The domain of f ′ is the set of points at which f has a derivative, so dom(f′)⊆dom(f) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiGacsgacaGGVbGaaiyBaiGacIcacaWGMbGaeyOmGiQaaiykaiabgAOinlGacsgacaGGVbGaaiyBaiaacIcacaWGMbGaaiykaaaa@4692@ .

The algebra of derivatives turns out to be pretty simple.

Example 1: The derivative of f(x)= x 2 −1 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamiEaiaacMcacqGH9aqpcaWG4bWdamaaCaaaleqabaWdbiaaikdaaaGccqGHsislcaaIXaaaaa@40E9@  at x=3 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGH9aqpcaaIZaaaaa@3BAB@  is given by

f′(3)= lim x→3 f(x)−f(3) x−3 = lim x→3 ( x 2 −1)−8 x−3 = lim x→3 x+3=6. MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@6EAF@

It is not much more difficult to compute the derivative at x=a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGH9aqpcaWGHbaaaa@3BD4@ :

f′(a)= lim x→a f(x)−f(a) x−a = lim x→a ( x 2 −1)−( a 2 −1) x−a = lim x→a x+a=2a. MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@7510@

This computation is valid for all real numbers a, so the function f′(x)=2x MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamiEaiaacMcacqGH9aqpcaaIYaGaamiEaaaa@406B@  is the derivative function of the function f(x)= x 2 −1 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamiEaiaacMcacqGH9aqpcaWG4bWdamaaCaaaleqabaWdbiaaikdaaaGccqGHsislcaaIXaaaaa@40E9@ .

Example 2: Let n∈ ℤ + MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaad6gacqGHiiIZcqWIKeIOpaWaaWbaaSqabeaapeGaey4kaScaaaaa@3E08@ , and let f(x)= x n MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamiEaiaacMcacqGH9aqpcaWG4bWdamaaCaaaleqabaWdbiaad6gaaaaaaa@3F6E@  for all x∈ℝ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcqWIDesOaaa@3CDC@ . Let a∈ℝ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadggacqGHiiIZcqWIDesOaaa@3CC5@ , then f(x)−f(a)= x n − a n =(x−a)( x n−1 +a x n−2 + a 2 x n−3 +…+ a n−2 x+ a n−1 ). MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@6B34@  Therefore,

f(x)−f(a) x−a = x n−1 +a x n−2 + a 2 x n−3 +…+ a n−2 x+ a n−1 , MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@6266@  for x≠a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHGjsUcaWGHbaaaa@3C95@ . Thus,

f′(a)= lim x→a f(x)−f(a) x−a = a n−1 +a a n−2 + a 2 a n−3 +…+ a n−2 a+ a n−1 =n a n−1 . MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@74F5@

 

Figure 2

We want to find the derivative of the function f(x)=sin(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamiEaiaacMcacqGH9aqpciGGZbGaaiyAaiaac6gacaGGOaGaamiEaiaacMcaaaa@4260@ .  This will require the following lemma.

Lemma 1: lim x→0 sinx x =1. MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaaIWaaapaqabaGcpeWaaSaaa8aabaWdbiGacohacaGGPbGaaiOBaiaadIhaa8aabaWdbiaadIhaaaGaeyypa0JaaGymaiaac6caaaa@4773@

Proof: In Figure 1 we are looking at a section of the unit circle. The area, A I MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadgeapaWaaSbaaSqaa8qacaWGjbaapaqabaaaaa@3AD9@ , of the inner triangle, ΔABD MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiabfs5aejaadgeacaWGcbGaamiraaaa@3CA7@ , has area 1 2 sinxcosx MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacaaIXaaapaqaa8qacaaIYaaaaiGacohacaGGPbGaaiOBaiaadIhaciGGJbGaai4BaiaacohacaWG4baaaa@4255@ . The outer triangle, ΔACE MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiabfs5aejaadgeacaWGdbGaamyraaaa@3CA9@ , has area A O = 1 2 tanx MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadgeapaWaaSbaaSqaa8qacaWGpbaapaqabaGcpeGaeyypa0ZaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaaGaciiDaiaacggacaGGUbGaamiEaaaa@4192@ .

The sector of the circle defined by A, C, and D has area

A S = x 2 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadgeapaWaaSbaaSqaa8qacaWGtbaapaqabaGcpeGaeyypa0ZaaSaaa8aabaWdbiaadIhaa8aabaWdbiaaikdaaaaaaa@3E0A@ . We clearly see that A I < A S < A O . MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadgeapaWaaSbaaSqaa8qacaWGjbaapaqabaGcpeGaeyipaWJaamyqa8aadaWgaaWcbaWdbiaadofaa8aabeaak8qacqGH8aapcaWGbbWdamaaBaaaleaapeGaam4taaWdaeqaaOWdbiaac6caaaa@41CD@   This means that

sinxcosx 2 < x 2 < tanx 2 sinxcosx< x<tanx cosx< x sinx < 1 cosx cosx< sinx x < 1 cosx MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@7989@

Thus,

lim x→0 cosx≤ lim x→0 sinx x ≤ lim x→0 1 cosx MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaaIWaaapaqabaGcpeGaci4yaiaac+gacaGGZbGaamiEaiabgsMiJ+aadaWfqaqaa8qaciGGSbGaaiyAaiaac2gaaSWdaeaapeGaamiEaiabgkziUkaaicdaa8aabeaak8qadaWcaaWdaeaapeGaci4CaiaacMgacaGGUbGaamiEaaWdaeaapeGaamiEaaaacqGHKjYOpaWaaCbeaeaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaaIWaaapaqabaGcpeWaaSaaa8aabaWdbiaaigdaa8aabaWdbiGacogacaGGVbGaai4CaiaadIhaaaaaaa@5F3B@  which leads to 1≤ lim x→0 sinx x ≤1. MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaaigdacqGHKjYOpaWaaCbeaeaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaaIWaaapaqabaGcpeWaaSaaa8aabaWdbiGacohacaGGPbGaaiOBaiaadIhaa8aabaWdbiaadIhaaaGaeyizImQaaGymaiaac6caaaa@4AB1@   Therefore, we have that lim x→0 sinx x =1. MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaaIWaaapaqabaGcpeWaaSaaa8aabaWdbiGacohacaGGPbGaaiOBaiaadIhaa8aabaWdbiaadIhaaaGaeyypa0JaaGymaiaac6caaaa@4773@   

Now, to find the derivative of the sine function we have:

lim x→a sinx−sina x−a = lim x→a 2sin( x−a 2 )cos( x+a 2 ) x−a = lim x→a sin( x−a 2 )cos( x+a 2 ) x−a 2 = lim x→a sin( x−a 2 ) x−a 2 cos( x+a 2 ) = lim u→0 sin(u) u lim u→0 cos(u+a) =1·cosa=cosa MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@C968@

Thus, we get that the derivative function for sin x is cos x.

We can show that the derivative function of the cosine function is – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@ sin x in a similar manner.

lim x→a cosx−cosa x−a = lim x→a −2sin( x+a 2 )sin( x−a 2 ) x−a = lim x→a − sin( x−a 2 ) x−a 2 sin( x+a 2 ) = lim x→a − sin( x−a 2 ) x−a 2 lim x→a sin( x+a 2 ) =−1·sina=−sina MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@B9CD@

One of the first things we should notice is that differentiability at a point implies continuity at a point.

Theorem 1: If f is differentiable at x = a, then f is continuous at x = a.

Proof: Since f is differentiable at x=a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGH9aqpcaWGHbaaaa@3BD4@ , we know that

f′(a)= lim x→a f(x)−f(a) x−a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamyyaiaacMcacqGH9aqppaWaaCbeaeaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGHbaapaqabaGcpeWaaSaaa8aabaWdbiaadAgacaGGOaGaamiEaiaacMcacqGHsislcaWGMbGaaiikaiaadggacaGGPaaapaqaa8qacaWG4bGaeyOeI0Iaamyyaaaaaaa@5051@

exists and is finite.  We need to show that lim x→a f(x)=f(a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGHbaapaqabaGcpeGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iaadAgacaGGOaGaamyyaiaacMcaaaa@477D@ . We have f(x)=(x−a) f(x)−f(a) x−a +f(a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamiEaiaacMcacqGH9aqpcaGGOaGaamiEaiabgkHiTiaadggacaGGPaWaaSaaa8aabaWdbiaadAgacaGGOaGaamiEaiaacMcacqGHsislcaWGMbGaaiikaiaadggacaGGPaaapaqaa8qacaWG4bGaeyOeI0IaamyyaaaacqGHRaWkcaWGMbGaaiikaiaadggacaGGPaaaaa@4FDD@  for all x≠a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHGjsUcaWGHbaaaa@3C95@  in the domain of f MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgaaaa@39D6@ . Taking the limit of both sides as x→a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHsgIRcaWGHbaaaa@3CBB@  we get

lim x→a f(x) = lim x→a [ (x−a) f(x)−f(a) x−a +f(a) ] = lim x→a [ (x−a) f(x)−f(a) x−a ]+ lim x→a f(a) = lim x→a (x−a) lim x→a f(x)−f(a) x−a +f(a) =0·f′(a)+f(a)=f(a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@B734@

We are done.  � MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaajugGbabaaaaaaaaapeGaeSy==7gaaa@3CD5@

Note that the other direction is not true.  What is a good counterexample?

V. Rules of Differentiation

We now want to look at the basic rules of differentiation.  We will start with the easy Sum and Difference Rules.

Theorem 2: If f and g are both differentiable, then f + g and f – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieGajugGbabaaaaaaaaapeGaa83eGaaa@3A99@  g are differentiable  and

(f±g)′(x)=f′(x)±g′(x). MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacaWGMbGaeyySaeRaam4zaiaacMcacqGHYaIOcaGGOaGaamiEaiaacMcacqGH9aqpcaWGMbGaeyOmGiQaaiikaiaadIhacaGGPaGaeyySaeRaam4zaiabgkdiIkaacIcacaWG4bGaaiykaiaac6caaaa@4F08@

Proof: We will show that (f+g)′(a)=f′(a)+g′(a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacaWGMbGaey4kaSIaam4zaiaacMcacqGHYaIOcaGGOaGaamyyaiaacMcacqGH9aqpcaWGMbGaeyOmGiQaaiikaiaadggacaGGPaGaey4kaSIaam4zaiabgkdiIkaacIcacaWGHbGaaiykaaaa@4BF9@  for an arbitrary a∈domf MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadggacqGHiiIZpaGaciizaiaac+gacaGGTbWdbiaadAgaaaa@3F2D@ .

lim x→a (f+g)(x)−(f+g)(a) x−a = lim x→a f(x)+g(x)−(f(a)+g(a)) x−a = lim x→a [ f(x)−f(a) x−a + g(x)−g(a) x−a ] =f′(a)+g′(a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@98E9@

The difference is proven in exactly the same way.  � MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaajugGbabaaaaaaaaapeGaeSy==7gaaa@3CD5@

This comes as no surprise.

Theorem 3: If c∈ℝ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadogacqGHiiIZcqWIDesOaaa@3CC7@  and f is the constant function given by f (x) = c for all x∈ℝ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcqWIDesOaaa@3CDC@ , then f′(x)=0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamiEaiaacMcacqGH9aqpcaaIWaaaaa@3F6C@  for all x∈ℝ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcqWIDesOaaa@3CDC@ .

Proof: Simply compute the derivative:

f′(a) = lim x→a f(x)−f(a) x−a = lim x→a 0 x−a =0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@5E7C@

and we are done.  � MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaajugGbabaaaaaaaaapeGaeSy==7gaaa@3CD5@

We need to compute three more general derivatives: the derivative of a product, the derivative of a quotient, and the derivative of a composition.

V.1    Product Rule

The student product rule would be what we would expect:

(fg)′(x)=f′(x)g′(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacaWGMbGaam4zaiaacMcacqGHYaIOcaGGOaGaamiEaiaacMcacqGH9aqpcaWGMbGaeyOmGiQaaiikaiaadIhacaGGPaGaam4zaiabgkdiIkaacIcacaWG4bGaaiykaaaa@4A7A@ .

This is not true.  Simple counterexamples are numerous.

The real problem lies in the difference quotient.  Note that

f(x)g(x)−f(a)g(a) x−a ≠ f(x)−f(a) x−a · g(x)−g(a) x−a . MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacaWGMbGaaiikaiaadIhacaGGPaGaam4zaiaacIcacaWG4bGaaiykaiabgkHiTiaadAgacaGGOaGaamyyaiaacMcacaWGNbGaaiikaiaadggacaGGPaaapaqaa8qacaWG4bGaeyOeI0IaamyyaaaacqGHGjsUdaWcaaWdaeaapeGaamOzaiaacIcacaWG4bGaaiykaiabgkHiTiaadAgacaGGOaGaamyyaiaacMcaa8aabaWdbiaadIhacqGHsislcaWGHbaaaiaacEladaWcaaWdaeaapeGaam4zaiaacIcacaWG4bGaaiykaiabgkHiTiaadEgacaGGOaGaamyyaiaacMcaa8aabaWdbiaadIhacqGHsislcaWGHbaaaiaac6caaaa@6270@

Instead, note that

f(x)g(x)−f(a)g(a) x−a = f(x)g(x)−f(x)g(a)+f(x)g(a)−f(a)g(a) x−a =f(x) g(x)−g(a) x−a +g(a) f(x)−f(a) x−a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@87ED@

Now, taking the limit as x→a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHsgIRcaWGHbaaaa@3CBB@  gives us that

(fg)′(x)=f′(x)g(x)+f(x)g′(x). MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacaWGMbGaam4zaiaacMcacqGHYaIOcaGGOaGaamiEaiaacMcacqGH9aqpcaWGMbGaeyOmGiQaaiikaiaadIhacaGGPaGaam4zaiaacIcacaWG4bGaaiykaiabgUcaRiaadAgacaGGOaGaamiEaiaacMcacaWGNbGaeyOmGiQaaiikaiaadIhacaGGPaGaaiOlaaaa@5291@

This is the Product Rule for derivatives.

Note that the following is an immediate corollary of the previous two rules.

Corollary 1: (cf)′(x)=c·f′(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacaWGJbGaamOzaiaacMcacqGHYaIOcaGGOaGaamiEaiaacMcacqGH9aqpcaWGJbGaai4TaiaadAgacqGHYaIOcaGGOaGaamiEaiaacMcaaaa@47D7@  if f MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgaaaa@39D6@  is differentiable and c∈ℝ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadogacqGHiiIZcqWIDesOaaa@3CC7@ .

 

V.2    Quotient Rule

Theorem 4: If f and g are differentiable at x = a and g′(a)≠0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgacqGHYaIOcaGGOaGaamyyaiaacMcacqGHGjsUcaaIWaaaaa@4017@ , then

( f g )′(a)= g(a)f′(a)−f(a)g′(a) (g(a)) 2 . MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbmaabmaapaqaa8qadaWcaaWdaeaapeGaamOzaaWdaeaapeGaam4zaaaaaiaawIcacaGLPaaacqGHYaIOcaGGOaGaamyyaiaacMcacqGH9aqpdaWcaaWdaeaapeGaam4zaiaacIcacaWGHbGaaiykaiaadAgacqGHYaIOcaGGOaGaamyyaiaacMcacqGHsislcaWGMbGaaiikaiaadggacaGGPaGaam4zaiabgkdiIkaacIcacaWGHbGaaiykaaWdaeaapeGaaiikaiaadEgacaGGOaGaamyyaiaacMcacaGGPaWdamaaCaaaleqabaWdbiaaikdaaaaaaOGaaiOlaaaa@58AA@

Proof: Again, we just have to rewrite the quotient appropriately. Since g′(a)≠0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgacqGHYaIOcaGGOaGaamyyaiaacMcacqGHGjsUcaaIWaaaaa@4017@  and g is continuous at x = a, there is an open interval I containing a so that g(x)≠0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgacaGGOaGaamiEaiaacMcacqGHGjsUcaaIWaaaaa@3EAE@  for x∈I MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaWGjbaaaa@3C3A@ . Thus, for x∈I MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaWGjbaaaa@3C3A@ , we can write

(f/g)(x)−(f/g)(a) = f(x) g(x) − f(a) g(a) = g(a)f(x)−f(a)g(x) g(x)g(a) = g(a)f(x)−g(a)f(a)+g(a)f(a)−f(a)g(x) g(x)g(a)    so  (f/g)(x)−(f/g)(a) x−a =[ g(a) f(x)−f(a) x−a −f(a) g(x)−g(a) x−a ] 1 g(x)g(a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@CB66@

Taking the limits as x→a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHsgIRcaWGHbaaaa@3CBB@  gives us the result.  � MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaajugGbabaaaaaaaaapeGaeSy==7gaaa@3CD5@

V.3    The Chain Rule

Theorem 5: If f is differentiable at a and g is differentiable at f (a), then the composite function g∘f MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgacqWIyiYBcaWGMbaaaa@3BFC@  is differentiable at a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadggaaaa@39D1@  and (g∘f)′(a)=g′(f(a))·f′(a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacaWGNbGaeSigI8MaamOzaiaacMcacqGHYaIOcaGGOaGaamyyaiaacMcacqGH9aqpcaWGNbGaeyOmGiQaaiikaiaadAgacaGGOaGaamyyaiaacMcacaGGPaGaai4TaiaadAgacqGHYaIOcaGGOaGaamyyaiaacMcaaaa@4EEE@ .

Proof: Since g is differentiable at f (a) it can be shown that g∘f MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgacqWIyiYBcaWGMbaaaa@3BFC@  is defined on some open interval containing a. Let

h(y)= g(y)−g(f(a)) y−f(a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIgacaGGOaGaamyEaiaacMcacqGH9aqpdaWcaaWdaeaapeGaam4zaiaacIcacaWG5bGaaiykaiabgkHiTiaadEgacaGGOaGaamOzaiaacIcacaWGHbGaaiykaiaacMcaa8aabaWdbiaadMhacqGHsislcaWGMbGaaiikaiaadggacaGGPaaaaaaa@4C37@

for y∈domg MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadMhacqGHiiIZpaGaciizaiaac+gacaGGTbWdbiaadEgaaaa@3F46@  and y≠f(a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadMhacqGHGjsUcaWGMbGaaiikaiaadggacaGGPaaaaa@3EDA@ , and let h(f(a))=g′(f(a)) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIgacaGGOaGaamOzaiaacIcacaWGHbGaaiykaiaacMcacqGH9aqpcaWGNbGaeyOmGiQaaiikaiaadAgacaGGOaGaamyyaiaacMcacaGGPaaaaa@4650@ .  Since lim y→f(a) h(y)=h(f(a)) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadMhacqGHsgIRcaWGMbGaaiikaiaadggacaGGPaaapaqabaGcpeGaamiAaiaacIcacaWG5bGaaiykaiabg2da9iaadIgacaGGOaGaamOzaiaacIcacaWGHbGaaiykaiaacMcaaaa@4C0B@ , the function h MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIgaaaa@39D8@  is continuous at f(a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamyyaiaacMcaaaa@3C15@ .

g(y)−g(f(a))=h(y)[y−f(a)] MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgacaGGOaGaamyEaiaacMcacqGHsislcaWGNbGaaiikaiaadAgacaGGOaGaamyyaiaacMcacaGGPaGaeyypa0JaamiAaiaacIcacaWG5bGaaiykaiaacUfacaWG5bGaeyOeI0IaamOzaiaacIcacaWGHbGaaiykaiaac2faaaa@4DA9@

for all y∈domg MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadMhacqGHiiIZcaqGKbGaae4Baiaab2gacaWGNbaaaa@3F22@  so

(g∘f)(x)−(g∘f)(a)=h(f(x))[f(x)−f(a)] for x∈dom(g∘f). MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacaWGNbGaeSigI8MaamOzaiaacMcacaGGOaGaamiEaiaacMcacqGHsislcaGGOaGaam4zaiablIHiVjaadAgacaGGPaGaaiikaiaadggacaGGPaGaeyypa0JaamiAaiaacIcacaWGMbGaaiikaiaadIhacaGGPaGaaiykaiaacUfacaWGMbGaaiikaiaadIhacaGGPaGaeyOeI0IaamOzaiaacIcacaWGHbGaaiykaiaac2facaqGGaGaaeOzaiaab+gacaqGYbGaaeiiaiaadIhacqGHiiIZcaqGKbGaae4Baiaab2gacaGGOaGaam4zaiablIHiVjaadAgacaGGPaGaaiOlaaaa@6562@

Therefore

(g∘f)(x)−(g∘f)(a) x−a =h(f(x)) f(x)−f(a) x−a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacaGGOaGaam4zaiablIHiVjaadAgacaGGPaGaaiikaiaadIhacaGGPaGaeyOeI0IaaiikaiaadEgacqWIyiYBcaWGMbGaaiykaiaacIcacaWGHbGaaiykaaWdaeaapeGaamiEaiabgkHiTiaadggaaaGaeyypa0JaamiAaiaacIcacaWGMbGaaiikaiaadIhacaGGPaGaaiykamaalaaapaqaa8qacaWGMbGaaiikaiaadIhacaGGPaGaeyOeI0IaamOzaiaacIcacaWGHbGaaiykaaWdaeaapeGaamiEaiabgkHiTiaadggaaaaaaa@5B62@

for x∈dom(g∘f) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaqGKbGaae4Baiaab2gacaGGOaGaam4zaiablIHiVjaadAgacaGGPaaaaa@429F@ , x≠a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHGjsUcaWGHbaaaa@3C95@ .  Since lim x→a f(x)=f(a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGHbaapaqabaGcpeGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iaadAgacaGGOaGaamyyaiaacMcaaaa@477D@  and the function h is continuous at
f (a), lim x→a h(f(x))=h(f(a))=g′(f(a)). MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGHbaapaqabaGcpeGaamiAaiaacIcacaWGMbGaaiikaiaadIhacaGGPaGaaiykaiabg2da9iaadIgacaGGOaGaamOzaiaacIcacaWGHbGaaiykaiaacMcacqGH9aqpcaWGNbGaeyOmGiQaaiikaiaadAgacaGGOaGaamyyaiaacMcacaGGPaGaaiOlaaaa@54B0@   The other limit in the above is f′(a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamyyaiaacMcaaaa@3D95@ , so the result follows.  � MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaajugGbabaaaaaaaaapeGaeSy==7gaaa@3CD5@

V.4    Other Transcendental Derivatives

We should find the derivatives of the natural logarithm and the exponential functions. However, for a strictly rigorous treatment we will defer until we do integration.  For now, we will deal with these functions as follows.

First, recall that

lim n→∞ ( 1+ 1 n ) n =e. MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaad6gacqGHsgIRcqGHEisPa8aabeaak8qadaqadaWdaeaapeGaaGymaiabgUcaRmaalaaapaqaa8qacaaIXaaapaqaa8qacaWGUbaaaaGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaWGUbaaaOGaeyypa0Jaamyzaiaac6caaaa@49B9@

Now, we want to compute the derivative of the natural logarithm, ln x which is defined by ln a = b means a = eb.  We know that this function satisfies all of the usual properties of the logarithms, so we begin with a lemma.

Lemma 2:  If f is differentiable at x = a, then

f′(a)= lim h→0 f(a+h)−f(a) h . MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamyyaiaacMcacqGH9aqppaWaaCbeaeaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIgacqGHsgIRcaaIWaaapaqabaGcpeWaaSaaa8aabaWdbiaadAgacaGGOaGaamyyaiabgUcaRiaadIgacaGGPaGaeyOeI0IaamOzaiaacIcacaWGHbGaaiykaaWdaeaapeGaamiAaaaacaGGUaaaaa@509C@

Then, if f (x) = ln x we have

f′(x) = lim h→0 ln(x+h)−lnx h = lim h→0 1 h ln( x+h x ) = lim h→0 1 h ln( 1+ h x ) = lim h→0 1 x x h ln( 1+ h x ) = lim h→0 1 x ln[ ( 1+ h x ) x/h ] = 1 x lim t→∞ ln ( 1+ 1 t ) t = 1 x ln lim t→∞ ( 1+ 1 t ) t = 1 x ln(e)= 1 x MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@CFDC@

Now, we made a number of assumptions here, such as the logarithm is a continuous function and others.  Nonetheless, we are okay for now.

We have that the derivative of the natural logarithm of x is 1/x.  Now, to the exponential function.  Here we will use the Chain Rule.

ln( e x ) =x d  dx ln( e x ) = d  dx x 1 e x d  dx e x =1 d  dx e x = e x MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@6835@

From these and the Chain Rule and the Product Rule the following are true:

d  dx sinx=cosx d  dx cosx=−sinx d  dx tanx= sec 2 x d  dx cotx=− csc 2 x d  dx secx=secxtanx d  dx cscx=−cscxcotx d  dx e x = e x d  dx a x = a x lna d  dx lnx= 1 x d  dx log b x= 1 xlnb d  dx sinhx=coshx d  dx coshx=sinhx MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@E63E@

Recall that we had defined the hyperbolic sine and cosine by

sinhx= e x − e −x 2     coshx= e x + e −x 2 . MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiGacohacaGGPbGaaiOBaiaacIgacaWG4bGaeyypa0ZaaSaaa8aabaWdbiaadwgapaWaaWbaaSqabeaapeGaamiEaaaakiabgkHiTiaadwgapaWaaWbaaSqabeaapeGaeyOeI0IaamiEaaaaaOWdaeaapeGaaGOmaaaacaaMg8UaaGPbVlGacogacaGGVbGaai4CaiaacIgacaWG4bGaeyypa0ZaaSaaa8aabaWdbiaadwgapaWaaWbaaSqabeaapeGaamiEaaaakiabgUcaRiaadwgapaWaaWbaaSqabeaapeGaeyOeI0IaamiEaaaaaOWdaeaapeGaaGOmaaaacaGGUaaaaa@56F5@

Recall that the Lambert W-function is defined by W(a) = b means that a=b· e b MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadggacqGH9aqpcaWGIbGaai4TaiaadwgapaWaaWbaaSqabeaapeGaamOyaaaaaaa@3F16@ . Thus, to find the derivative of this function we will use the Product Rule and the Chain Rule.

x=W(x) e W(x) dx dx = d  dx ( W(x) e W(x) ) 1=W′(x) e W(x) +W(x) e W(x) W′(x) W′(x)= 1 e W(x) (1+W(x)) but  e W(x) =x/W(x), so W′(x)= W(x) x(1+W(x)) , x≠0,−1/e MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakqaaeeqaaiaadIhacqGH9aqpqaaaaaaaaaWdbiaadEfacaGGOaGaamiEaiaacMcacaWGLbWdamaaCaaaleqabaWdbiaadEfacaGGOaGaamiEaiaacMcaaaaak8aabaWdbmaalaaapaqaa8qacaWGKbGaamiEaaWdaeaapeGaamizaiaadIhaaaGaeyypa0ZaaSaaa8aabaWdbiaadsgacaaMe8oapaqaa8qacaWGKbGaamiEaaaadaqadaWdaeaapeGaam4vaiaacIcacaWG4bGaaiykaiaadwgapaWaaWbaaSqabeaapeGaam4vaiaacIcacaWG4bGaaiykaaaaaOGaayjkaiaawMcaaaqaaiaaigdacqGH9aqpcaWGxbGaeyOmGiQaaiikaiaadIhacaGGPaGaamyza8aadaahaaWcbeqaa8qacaWGxbGaaiikaiaadIhacaGGPaaaaOGaey4kaSIaam4vaiaacIcacaWG4bGaaiykaiaadwgapaWaaWbaaSqabeaapeGaam4vaiaacIcacaWG4bGaaiykaaaakiaadEfacqGHYaIOcaGGOaGaamiEaiaacMcaaeaacaWGxbGaeyOmGiQaaiikaiaadIhacaGGPaGaeyypa0ZaaSaaa8aabaWdbiaaigdaa8aabaWdbiaadwgapaWaaWbaaSqabeaapeGaam4vaiaacIcacaWG4bGaaiykaaaakiaacIcacaaIXaGaey4kaSIaam4vaiaacIcacaWG4bGaaiykaiaacMcaaaaabaGaaeOyaiaabwhacaqG0bGaaeiiaiaadwgapaWaaWbaaSqabeaapeGaam4vaiaacIcacaWG4bGaaiykaaaakiabg2da9iaadIhacaGGVaGaam4vaiaacIcacaWG4bGaaiykaiaacYcacaqGGaGaae4Caiaab+gaaeaacaWGxbGaeyOmGiQaaiikaiaadIhacaGGPaGaeyypa0ZaaSaaa8aabaWdbiaadEfacaGGOaGaamiEaiaacMcaa8aabaWdbiaadIhacaGGOaGaaGymaiabgUcaRiaadEfacaGGOaGaamiEaiaacMcacaGGPaaaaiaacYcacaaMe8UaamiEaiabgcMi5kaaicdacaGGSaGaeyOeI0IaaGymaiaac+cacaWGLbaaaaa@AA66@

There are very few functions that we cannot differentiate using these theorems, if they have a derivative.

This technique leads us to a more general theorem.

Theorem 6: Let f be a continuous one-to-one function defined on an interval and suppose that f is differentiable at a= f −1 (b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadggacqGH9aqpcaWGMbWdamaaCaaaleqabaWdbiabgkHiTiaaigdaaaGccaGGOaGaamOyaiaacMcaaaa@4000@ , with f′(a)≠0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamyyaiaacMcacqGHGjsUcaaIWaaaaa@4016@ , then f −1 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgapaWaaWbaaSqabeaapeGaeyOeI0IaaGymaaaaaaa@3BCA@  is differentiable at b and

( f −1 ) ′ (b)= 1 f′( f −1 (b)) . MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacaWGMbWdamaaCaaaleqabaWdbiabgkHiTiaaigdaaaGcceGGPaGbauaacaGGOaGaamOyaiaacMcacqGH9aqpdaWcaaWdaeaapeGaaGymaaWdaeaapeGaamOzaiabgkdiIkaacIcacaWGMbWdamaaCaaaleqabaWdbiabgkHiTiaaigdaaaGccaGGOaGaamOyaiaacMcacaGGPaaaaiaac6caaaa@4B27@

Proof: Let b = f (a).  Then

lim h→0 f −1 (b+h)− f −1 (b) h = lim h→0 f −1 (b+h)−a h . MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@60DB@

Now, every number b+h in the domain of f -1 can be written in the form b + h = f (a + k) for a unique k.  Then

lim h→0 f −1 (b+h)−a h = lim h→0 f −1 (f(a+k))−a f(a+k)−b = lim h→0 k f(a+k)−f(a) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@76B7@

Now, since b + h = f(a+k) we have

b+h =f(a+k) f −1 (b+h) =a+k k = f −1 (b+h)−a= f −1 (b+h)− f −1 (b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@65DB@

The function f −1 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgapaWaaWbaaSqabeaapeGaeyOeI0IaaGymaaaaaaa@3BCA@  is continuous at b, so we have that k approaches 0 as h approaches 0.  Since

lim k→0 f(a+k)−f(a) k =f′(a)=f′( f −1 (b))≠0, MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadUgacqGHsgIRcaaIWaaapaqabaGcpeWaaSaaa8aabaWdbiaadAgacaGGOaGaamyyaiabgUcaRiaadUgacaGGPaGaeyOeI0IaamOzaiaacIcacaWGHbGaaiykaaWdaeaapeGaam4AaaaacqGH9aqpcaWGMbGaeyOmGiQaaiikaiaadggacaGGPaGaeyypa0JaamOzaiabgkdiIkaacIcacaWGMbWdamaaCaaaleqabaWdbiabgkHiTiaaigdaaaGccaGGOaGaamOyaiaacMcacaGGPaGaeyiyIKRaaGimaiaacYcaaaa@5CF8@

we have that

( f −1 )′(b)= 1 f′( f −1 (b)) . MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacaWGMbWdamaaCaaaleqabaWdbiabgkHiTiaaigdaaaGccaGGPaGaeyOmGiQaaiikaiaadkgacaGGPaGaeyypa0ZaaSaaa8aabaWdbiaaigdaa8aabaWdbiaadAgacqGHYaIOcaGGOaGaamOza8aadaahaaWcbeqaa8qacqGHsislcaaIXaaaaOGaaiikaiaadkgacaGGPaGaaiykaaaacaGGUaaaaa@4C9B@   � MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaajugGbabaaaaaaaaapeGaeSy==7gaaa@3CD5@

This leads to the derivatives of the inverse trigonometric functions.  Let’s consider the arctangent function.  First, we will choose the tangent with domain (−π/2,π/2) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacqGHsislcqaHapaCcaGGVaGaaGOmaiaacYcacqaHapaCcaGGVaGaaGOmaiaacMcaaaa@4239@ . The range is ℝ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiabl2riHcaa@3A5B@  and the arctangent will have domain ℝ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiabl2riHcaa@3A5B@  and range (−π/2,π/2) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacqGHsislcqaHapaCcaGGVaGaaGOmaiaacYcacqaHapaCcaGGVaGaaGOmaiaacMcaaaa@4239@ . I will use the notation arctan for the arctangent function.

tan(arctanx)=x sec 2 (arctanx) d  dx arctanx=1 d  dx arctanx= cos 2 (arctanx) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@7680@

However, because these are trigonometric functions, we know more. The arctangent of x is the angle whose tangent is x.  The tangent of that angle can be found from a right triangle whose ratio of the opposite side to the adjacent side is x:1.  Thus, we can take the opposite side to have length x and the adjacent side to have length 1.  That means that the hypotenuse is 1+ x 2 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbmaakaaapaqaa8qacaaIXaGaey4kaSIaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaaqabaaaaa@3CBC@ . Then the cosine of that angle is the ratio of the adjacent side to the hypotenuse, or 1: 1+ x 2 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaaigdacaGG6aWaaOaaa8aabaWdbiaaigdacqGHRaWkcaWG4bWdamaaCaaaleqabaWdbiaaikdaaaaabeaaaaa@3E35@ . Thus,

cos(arctan(x))= 1 1+ x 2 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiGacogacaGGVbGaai4CaiaacIcaciGGHbGaaiOCaiaacogacaGG0bGaaiyyaiaac6gacaGGOaGaamiEaiaacMcacaGGPaGaeyypa0ZaaSaaa8aabaWdbiaaigdaa8aabaWdbmaakaaapaqaa8qacaaIXaGaey4kaSIaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaaqabaaaaaaa@4AE0@

and therefore

d  dx arctanx= 1 1+ x 2 . MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacaWGKbGaaGjbVdWdaeaapeGaamizaiaadIhaaaGaciyyaiaackhacaGGJbGaaiiDaiaacggacaGGUbGaamiEaiabg2da9maalaaapaqaa8qacaaIXaaapaqaa8qacaaIXaGaey4kaSIaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaaaakiaac6caaaa@4A92@

Similar techniques show that

d  dx arcsinx= 1 1− x 2 d  dx arcsecx= 1 x x 2 −1 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@5C7B@

VI     The Mean Value Theorem}

We know that a continuous function on a closed interval must take on its maximum and minimum values.  How do we find these points?  This is one of the places where the tool of calculus is helpful.

Theorem 7: Let f is defined on an open interval containing c.  If f assumes its maximum or minimum value at x = c, and if f is differentiable at x = c, then f′(c)=0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH9aqpcaaIWaaaaa@3F57@ .

Proof: Suppose the f is defined on (a,b) with a < c < b.  Since either f or – MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaieaajugGbabaaaaaaaaapeGaa83eGaaa@3A97@ f  assumes its maximum at x = c, we may assume that f assumes its maximum at x = c.  We need to show that f′(c)=0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH9aqpcaaIWaaaaa@3F57@ .

Assume that f′(c)>0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH+aGpcaaIWaaaaa@3F59@ . Since

f′(c)= lim x→c f(x)−f(c) x−c >0, MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH9aqppaWaaCbeaeaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGJbaapaqabaGcpeWaaSaaa8aabaWdbiaadAgacaGGOaGaamiEaiaacMcacqGHsislcaWGMbGaaiikaiaadogacaGGPaaapaqaa8qacaWG4bGaeyOeI0Iaam4yaaaacqGH+aGpcaaIWaGaaiilaaaa@52CB@

there is a δ>0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiabes7aKjabg6da+iaaicdaaaa@3C52@  so that a<c−δ<c+δ<b MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadggacqGH8aapcaWGJbGaeyOeI0IaeqiTdqMaeyipaWJaam4yaiabgUcaRiabes7aKjabgYda8iaadkgaaaa@44AD@  and if 0<|x−c|<δ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaaicdacqGH8aapcaGG8bGaamiEaiabgkHiTiaadogacaGG8bGaeyipaWJaeqiTdqgaaa@4224@  then f(x)−f(c) x−c >0. MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacaWGMbGaaiikaiaadIhacaGGPaGaeyOeI0IaamOzaiaacIcacaWGJbGaaiykaaWdaeaapeGaamiEaiabgkHiTiaadogaaaGaeyOpa4JaaGimaiaac6caaaa@45D9@

If we choose an x∈(c,c+δ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaGGOaGaam4yaiaacYcacaWGJbGaey4kaSIaeqiTdqMaaiykaaaa@41CC@ , then from the above inequality we have that f(x)>f(c) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamiEaiaacMcacqGH+aGpcaWGMbGaaiikaiaadogacaGGPaaaaa@4060@ , contrary to the assumption that f has its maximum at x = c.  Likewise, if we assume that f′(c)<0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH8aapcaaIWaaaaa@3F55@ , there is a δ>0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiabes7aKjabg6da+iaaicdaaaa@3C52@  so that if 0<|x−c|<δ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaaicdacqGH8aapcaGG8bGaamiEaiabgkHiTiaadogacaGG8bGaeyipaWJaeqiTdqgaaa@4224@  then

f(x)−f(c) x−c <0. MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacaWGMbGaaiikaiaadIhacaGGPaGaeyOeI0IaamOzaiaacIcacaWGJbGaaiykaaWdaeaapeGaamiEaiabgkHiTiaadogaaaGaeyipaWJaaGimaiaac6caaaa@45D5@

If we choose an x∈(c−δ,c) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaGGOaGaam4yaiabgkHiTiabes7aKjaacYcacaWGJbGaaiykaaaa@41D7@ , then from the above inequality we have that f(x)>f(c) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamiEaiaacMcacqGH+aGpcaWGMbGaaiikaiaadogacaGGPaaaaa@4060@ , contrary to the assumption that f has its maximum at x = c.  Thus, we must have that f′(c)=0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH9aqpcaaIWaaaaa@3F57@ .  � MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaajugGbabaaaaaaaaapeGaeSy==7gaaa@3CD5@

Rolle’s Theorem: Let f be continuous on [a,b] that is differentiable on (a,b) and satisfies f (a) = f (b).  Then there exists at least one c∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadogacqGHiiIZcaGGOaGaamyyaiaacYcacaWGIbGaaiykaaaa@3F2D@  so that f′(c)=0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH9aqpcaaIWaaaaa@3F57@ .

Proof:  We know that f takes its maximum and its minimum in the interval [a,b].  Thus, there exist points x 0 , y 0 ∈[a,b] MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaaiilaiaadMhapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaeyicI4Saai4waiaadggacaGGSaGaamOyaiaac2faaaa@43B3@  so that f( x 0 )≤f(x)≤f( y 0 ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamiEa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGPaGaeyizImQaamOzaiaacIcacaWG4bGaaiykaiabgsMiJkaadAgacaGGOaGaamyEa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGPaaaaa@4875@  for all x∈[a,b] MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaGGBbGaamyyaiaacYcacaWGIbGaaiyxaaaa@3FA9@ . If x 0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhapaWaaSbaaSqaa8qacaaIWaaapaqabaaaaa@3AFC@  and y 0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadMhapaWaaSbaaSqaa8qacaaIWaaapaqabaaaaa@3AFD@  are both endpoints of [a,b], then f MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgaaaa@39D6@  must be a constant function since f(a)=f(b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamyyaiaacMcacqGH9aqpcaWGMbGaaiikaiaadkgacaGGPaaaaa@4046@  and f′(x)=0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamiEaiaacMcacqGH9aqpcaaIWaaaaa@3F6C@  for all x∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaGGOaGaamyyaiaacYcacaWGIbGaaiykaaaa@3F42@ . Otherwise, f MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgaaaa@39D6@  assumes either a maximum or a minimum at some point in the interior, (a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacaWGHbGaaiilaiaadkgacaGGPaaaaa@3CC1@ , and by the above Theorem, f′(x)=0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamiEaiaacMcacqGH9aqpcaaIWaaaaa@3F6C@ .  � MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaajugGbabaaaaaaaaapeGaeSy==7gaaa@3CD5@

This next theorem is a bit of a generalization of Rolle’s Theorem.  We can think of Rolle’s Theorem as saying that somewhere in (a,b) there is a point at which the function has the same slope as the slope of the secant through (a,f(a)) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacaWGHbGaaiilaiaadAgacaGGOaGaamyyaiaacMcacaGGPaaaaa@3F04@  and (b,f(b)) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacaWGIbGaaiilaiaadAgacaGGOaGaamOyaiaacMcacaGGPaaaaa@3F06@ , which is 0.

Mean Value Theorem: Let f be a continuous function on [a,b] that is differentiable on (a,b).  Then there is a point c∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadogacqGHiiIZcaGGOaGaamyyaiaacYcacaWGIbGaaiykaaaa@3F2D@  such that f′(c)= f(b)−f(a) b−a . MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH9aqpdaWcaaWdaeaapeGaamOzaiaacIcacaWGIbGaaiykaiabgkHiTiaadAgacaGGOaGaamyyaiaacMcaa8aabaWdbiaadkgacqGHsislcaWGHbaaaiaac6caaaa@4999@

Proof: Let h be the function whose graph is the secant line through (a,f(a)) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacaWGHbGaaiilaiaadAgacaGGOaGaamyyaiaacMcacaGGPaaaaa@3F04@  and (b,f(b)) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacaWGIbGaaiilaiaadAgacaGGOaGaamOyaiaacMcacaGGPaaaaa@3F06@ . Then we have that

h(a)=h(a), h(b)=h(b), h′(x)= f(b)−f(a) b−a . MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIgacaGGOaGaamyyaiaacMcacqGH9aqpcaWGObGaaiikaiaadggacaGGPaGaaiilaiaaywW7caWGObGaaiikaiaadkgacaGGPaGaeyypa0JaamiAaiaacIcacaWGIbGaaiykaiaacYcacaaMf8UaamiAaiabgkdiIkaacIcacaWG4bGaaiykaiabg2da9maalaaapaqaa8qacaWGMbGaaiikaiaadkgacaGGPaGaeyOeI0IaamOzaiaacIcacaWGHbGaaiykaaWdaeaapeGaamOyaiabgkHiTiaadggaaaGaaiOlaaaa@5CEA@

Let g(x)=f(x)−L(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgacaGGOaGaamiEaiaacMcacqGH9aqpcaWGMbGaaiikaiaadIhacaGGPaGaeyOeI0IaamitaiaacIcacaWG4bGaaiykaaaa@4488@  for all x∈[a,b] MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaGGBbGaamyyaiaacYcacaWGIbGaaiyxaaaa@3FA9@ . Then, g is continuous on [a,b] and differentiable on (a,b).  Note that g(a) = 0 = g(b).  Thus by Rolle’s Theorem there is a point c∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadogacqGHiiIZcaGGOaGaamyyaiaacYcacaWGIbGaaiykaaaa@3F2D@  so that g′(c)=0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH9aqpcaaIWaaaaa@3F58@ . But, if g′(c)=0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH9aqpcaaIWaaaaa@3F58@ , then f′(c)= L ′ (c)= f(b)−f(a) b−a . MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH9aqpceWGmbGbauaacaGGOaGaam4yaiaacMcacqGH9aqpdaWcaaWdaeaapeGaamOzaiaacIcacaWGIbGaaiykaiabgkHiTiaadAgacaGGOaGaamyyaiaacMcaa8aabaWdbiaadkgacqGHsislcaWGHbaaaiaac6caaaa@4DBD@    � MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaajugGbabaaaaaaaaapeGaeSy==7gaaa@3CD5@

Corollary 1:  Let f be a differentiable function on (a,b) such that f′(x)=0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamiEaiaacMcacqGH9aqpcaaIWaaaaa@3F6C@  for all x∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaGGOaGaamyyaiaacYcacaWGIbGaaiykaaaa@3F42@ . Then f is a constant function on (a,b).

Proof: If f is not constant, there exist x 0 , x 1 ∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaaiilaiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyicI4SaaiikaiaadggacaGGSaGaamOyaiaacMcaaaa@434C@  so that a< x 0 < x 1 <b MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadggacqGH8aapcaWG4bWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabgYda8iaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyipaWJaamOyaaaa@421B@  and f( x 0 )≠f( x 1 ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamiEa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGPaGaeyiyIKRaamOzaiaacIcacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaacMcaaaa@4391@ . By the Mean Value Theorem, for some c∈( x 0 , x 1 ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadogacqGHiiIZcaGGOaGaamiEa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGSaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGPaaaaa@41B7@  we have that

f′(c)= f( x 1 )−f( x 0 ) x 1 − x 0 ≠0, MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH9aqpdaWcaaWdaeaapeGaamOzaiaacIcacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaacMcacqGHsislcaWGMbGaaiikaiaadIhapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaaiykaaWdaeaapeGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGHsislcaWG4bWdamaaBaaaleaapeGaaGimaaWdaeqaaaaak8qacqGHGjsUcaaIWaGaaiilaaaa@512C@

which is a contradiction.  � MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaajugGbabaaaaaaaaapeGaeSy==7gaaa@3CD5@

Corollary 2:  Let f and g be differentiable functions on (a,b) such that f′=g′ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcqGH9aqpcaWGNbGaeyOmGikaaa@3EC8@  for all x∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaGGOaGaamyyaiaacYcacaWGIbGaaiykaaaa@3F42@ . Then there is a constant C such that f (x) = g(x) + C for all x∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaGGOaGaamyyaiaacYcacaWGIbGaaiykaaaa@3F42@ .

Proof:  Left to the reader.

This is a very important result for integral calculus because it guarantees that all antiderivatives for a given function differ by at most a constant. Thus, when you have found one antiderivative, all the others will look like that one plus a constant.  Note that the constant is not always explicit.

∫ sin xcosx dx = 1 2 sin 2 x+C by substitution u=sinx = 1 2 ∫ sin 2xdx using the trig identity sin2x=2sinxcosx =− 1 4 cos(2x)+C MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@9F89@

Your job is to see how those two are the same.

Let f be a real-valued function defined on an interval I.  We say that f is strictly increasing on I if x 1 , x 2 ∈I MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyicI4Saamysaaaa@4046@  and x 1 < x 2 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyipaWJaamiEa8aadaWgaaWcbaWdbiaaikdaa8aabeaaaaa@3E2E@  then f( x 1 )<f( x 2 ); MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGPaGaeyipaWJaamOzaiaacIcacaWG4bWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaacMcacaGG7aaaaa@438F@  strictly decreasing on I if x 1 , x 2 ∈I MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyicI4Saamysaaaa@4046@  and x 1 < x 2 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyipaWJaamiEa8aadaWgaaWcbaWdbiaaikdaa8aabeaaaaa@3E2E@  then f( x 1 )>f( x 2 ); MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGPaGaeyOpa4JaamOzaiaacIcacaWG4bWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaacMcacaGG7aaaaa@4393@  increasing on I if x 1 , x 2 ∈I MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyicI4Saamysaaaa@4046@  and x 1 < x 2 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyipaWJaamiEa8aadaWgaaWcbaWdbiaaikdaa8aabeaaaaa@3E2E@  then f( x 1 )≤f( x 2 ); MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGPaGaeyizImQaamOzaiaacIcacaWG4bWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaacMcacaGG7aaaaa@4440@  and decreasing on I if x 1 , x 2 ∈I and  x 1 < x 2  then f( x 1 )≥f( x 2 ). MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaadIhapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyicI4SaamysaiaabccacaqGHbGaaeOBaiaabsgacaqGGaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH8aapcaWG4bWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaabccacaqG0bGaaeiAaiaabwgacaqGUbGaaeiiaiaadAgacaGGOaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGPaGaeyyzImRaamOzaiaacIcacaWG4bWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaacMcacaGGUaaaaa@59FF@

Corollary 1: Let f be a differentiable function on an interval (a,b). Then

        i.            f is strictly increasing if f′(x)>0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamiEaiaacMcacqGH+aGpcaaIWaaaaa@3F6E@  for all x∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaGGOaGaamyyaiaacYcacaWGIbGaaiykaaaa@3F42@ ;

     ii.            f is strictly decreasing if f′(x)<0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamiEaiaacMcacqGH8aapcaaIWaaaaa@3F6A@  for all x∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaGGOaGaamyyaiaacYcacaWGIbGaaiykaaaa@3F42@ ;

   iii.            f is increasing if f′(x)≥0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamiEaiaacMcacqGHLjYScaaIWaaaaa@402C@  for all x∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaGGOaGaamyyaiaacYcacaWGIbGaaiykaaaa@3F42@ ;

   iv.            f is decreasing if f′(x)≤0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamiEaiaacMcacqGHKjYOcaaIWaaaaa@401B@  for all x∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaGGOaGaamyyaiaacYcacaWGIbGaaiykaaaa@3F42@ .

Proof: We will show the proof for the first, as the others are proven similarly.

Consider x 0 , x 1 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaaiilaiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaaaaa@3DD8@  where a< x 0 < x 1 <b MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadggacqGH8aapcaWG4bWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabgYda8iaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyipaWJaamOyaaaa@421B@ . By the Mean Value Theorem, for some c∈( x 0 , x 1 ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadogacqGHiiIZcaGGOaGaamiEa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGSaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGPaaaaa@41B7@  we have

f( x 1 )−f( x 0 ) x 1 − x 0 =f′(c)>0. MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacaWGMbGaaiikaiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiykaiabgkHiTiaadAgacaGGOaGaamiEa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGPaaapaqaa8qacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabgkHiTiaadIhapaWaaSbaaSqaa8qacaaIWaaapaqabaaaaOWdbiabg2da9iaadAgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH+aGpcaaIWaGaaiOlaaaa@506F@

Since x 1 − x 0 >0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyOeI0IaamiEa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqGH+aGpcaaIWaaaaa@3FF1@ , we have that f( x 1 )−f( x 0 )>0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGPaGaeyOeI0IaamOzaiaacIcacaWG4bWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiaacMcacqGH+aGpcaaIWaaaaa@4479@  or f( x 1 )>f( x 0 ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacaGGOaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGPaGaeyOpa4JaamOzaiaacIcacaWG4bWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiaacMcaaaa@42D2@ .  � MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaajugGbabaaaaaaaaapeGaeSy==7gaaa@3CD5@

Intermediate Value Theorem for Derivatives: Let f be a differentiable function on (a,b). Whenever a< x 0 < x 1 <b MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadggacqGH8aapcaWG4bWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabgYda8iaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyipaWJaamOyaaaa@421B@  and m MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaad2gaaaa@39DD@  lies between f′( x 0 ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamiEa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGPaaaaa@3EDA@  and f′( x 1 ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGPaaaaa@3EDB@ , then there is a c∈( x 1 , x 2 ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadogacqGHiiIZcaGGOaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGaamiEa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacaGGPaaaaa@41B9@  so that f′(c)=m MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH9aqpcaWGTbaaaa@3F8F@ .

Proof: Without loss of generality we may assume that f′( x 0 )<m<f′( x 1 ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaamiEa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGPaGaeyipaWJaamyBaiabgYda8iaadAgacqGHYaIOcaGGOaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGPaaaaa@47C4@ . Let g(x)=f(x)−mx MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgacaGGOaGaamiEaiaacMcacqGH9aqpcaWGMbGaaiikaiaadIhacaGGPaGaeyOeI0IaamyBaiaadIhaaaa@4350@  for x∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhacqGHiiIZcaGGOaGaamyyaiaacYcacaWGIbGaaiykaaaa@3F42@ . Then g′( x 0 )<0<g′( x 1 ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgacqGHYaIOcaGGOaGaamiEa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGPaGaeyipaWJaaGimaiabgYda8iaadEgacqGHYaIOcaGGOaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGPaaaaa@478E@ . We know that g MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgaaaa@39D7@  assumes its minimum on [ x 0 , x 1 ] MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacUfacaWG4bWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiaacYcacaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaac2faaaa@3FB2@  at some point c∈[ x 0 , x 1 ] MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadogacqGHiiIZcaGGBbGaamiEa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGSaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGDbaaaa@421E@ . Since g′( x 0 )= lim x→ x 0 g(x)−g( x 0 ) x− x 0 <0, MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgacqGHYaIOcaGGOaGaamiEa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGPaGaeyypa0ZdamaaxababaWdbiGacYgacaGGPbGaaiyBaaWcpaqaa8qacaWG4bGaeyOKH4QaamiEa8aadaWgaaadbaWdbiaaicdaa8aabeaaaSqabaGcpeWaaSaaa8aabaWdbiaadEgacaGGOaGaamiEaiaacMcacqGHsislcaWGNbGaaiikaiaadIhapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaaiykaaWdaeaapeGaamiEaiabgkHiTiaadIhapaWaaSbaaSqaa8qacaaIWaaapaqabaaaaOWdbiabgYda8iaaicdacaGGSaaaaa@57B9@   g(x)−g( x 0 )<0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgacaGGOaGaamiEaiaacMcacqGHsislcaWGNbGaaiikaiaadIhapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaaiykaiabgYda8iaaicdaaaa@4348@  for x MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhaaaa@39E8@  close to and larger than x 0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhapaWaaSbaaSqaa8qacaaIWaaapaqabaaaaa@3AFC@ . In particular, there exists y 0 ∈( x 0 , x 1 ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadMhapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaeyicI4SaaiikaiaadIhapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaaiilaiaadIhapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiykaaaa@42FB@  so that g( y 0 )<g( x 0 ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgacaGGOaGaamyEa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGPaGaeyipaWJaam4zaiaacIcacaWG4bWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiaacMcaaaa@42D0@ . Therefore g MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgaaaa@39D7@  does not assume its minimum at x 0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIhapaWaaSbaaSqaa8qacaaIWaaapaqabaaaaa@3AFC@ , so we must have that c≠ x 0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadogacqGHGjsUcaWG4bWdamaaBaaaleaapeGaaGimaaWdaeqaaaaa@3DAB@ . Likewise, we can show that c≠ x 1 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadogacqGHGjsUcaWG4bWdamaaBaaaleaapeGaaGymaaWdaeqaaaaa@3DAC@ , so that we have c∈( x 0 , x 1 ) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadogacqGHiiIZcaGGOaGaamiEa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGSaGaamiEa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGPaaaaa@41B7@  and g′(c)=0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadEgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH9aqpcaaIWaaaaa@3F58@  by our previous theorem. Therefore, f′(c)=g′(c)+m=m MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH9aqpcaWGNbGaeyOmGiQaaiikaiaadogacaGGPaGaey4kaSIaamyBaiabg2da9iaad2gaaaa@4716@ .  � MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaajugGbabaaaaaaaaapeGaeSy==7gaaa@3CD5@

VII.   l’Hospital’s Rule}

When we are computing limits, we often have to compute limits of the form

lim x→M f(x) g(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGnbaapaqabaGcpeWaaSaaa8aabaWdbiaadAgacaGGOaGaamiEaiaacMcaa8aabaWdbiaadEgacaGGOaGaamiEaiaacMcaaaaaaa@46C9@

where M can signify a MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadggaaaa@39D1@ , a + MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadggapaWaaWbaaSqabeaapeGaey4kaScaaaaa@3AFF@ , a − MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadggapaWaaWbaaSqabeaapeGaeyOeI0caaaaa@3B0A@ , +∞ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiabgUcaRiabg6HiLcaa@3B3E@ , or −∞ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiabgkHiTiabg6HiLcaa@3B49@ . We know from our definition that the limit exists and

lim x→M f(x) g(x) = lim x→M f(x) lim x→M g(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGnbaapaqabaGcpeWaaSaaa8aabaWdbiaadAgacaGGOaGaamiEaiaacMcaa8aabaWdbiaadEgacaGGOaGaamiEaiaacMcaaaGaeyypa0ZaaSaaa8aabaWaaCbeaeaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGnbaapaqabaGcpeGaamOzaiaacIcacaWG4bGaaiykaaWdaeaadaWfqaqaa8qaciGGSbGaaiyAaiaac2gaaSWdaeaapeGaamiEaiabgkziUkaad2eaa8aabeaak8qacaWGNbGaaiikaiaadIhacaGGPaaaaaaa@5CBA@

provided that lim x→M f(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGnbaapaqabaGcpeGaamOzaiaacIcacaWG4bGaaiykaaaa@4339@  and lim x→M g(x) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGnbaapaqabaGcpeGaam4zaiaacIcacaWG4bGaaiykaaaa@433A@  exist and are finite and provided that lim x→M g(x)≠0. MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaadaWfqaqaaabaaaaaaaaapeGaciiBaiaacMgacaGGTbaal8aabaWdbiaadIhacqGHsgIRcaWGnbaapaqabaGcpeGaam4zaiaacIcacaWG4bGaaiykaiabgcMi5kaaicdacaGGUaaaaa@466D@   There are cases though where the definition does not help.  When we arrive at an indeterminate form such as 0 0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacaaIWaaapaqaa8qacaaIWaaaaaaa@3AAD@  or ∞ ∞ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacqGHEisPa8aabaWdbiabg6HiLcaaaaa@3C1B@ , then we can apply l’Hospital’s Rule to help us try to find the limit. Moreover, other indeterminate forms, such as ∞−∞ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiabg6HiLkabgkHiTiabg6HiLcaa@3CBA@ , 1 ∞ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaaigdapaWaaWbaaSqabeaapeGaeyOhIukaaaaa@3B63@ , ∞ 0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiabg6HiL+aadaahaaWcbeqaa8qacaaIWaaaaaaa@3B62@ , 0 0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaaicdapaWaaWbaaSqabeaapeGaaGimaaaaaaa@3AAB@  or 0×∞ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaaicdacqGHxdaTcqGHEisPaaa@3D2D@ , can usually be reformulated so as to take the form 0 0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacaaIWaaapaqaa8qacaaIWaaaaaaa@3AAD@  or ∞ ∞ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbmaalaaapaqaa8qacqGHEisPa8aabaWdbiabg6HiLcaaaaa@3C1B@ .

We need a few tools to prove l’Hospital’s Rule.  So first we prove the Generalized Mean Value Theorem.

Generalized Mean Value Theorem:  Let f and g be continuous functions on [a,b] that are differentiable on (a,b).  Then there exists at least one c∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadogacqGHiiIZcaGGOaGaamyyaiaacYcacaWGIbGaaiykaaaa@3F2D@  such that

f′(c)[g(b)−g(a)]=g′(c)[f(b)−f(a)]. MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadAgacqGHYaIOcaGGOaGaam4yaiaacMcacaGGBbGaam4zaiaacIcacaWGIbGaaiykaiabgkHiTiaadEgacaGGOaGaamyyaiaacMcacaGGDbGaeyypa0Jaam4zaiabgkdiIkaacIcacaWGJbGaaiykaiaacUfacaWGMbGaaiikaiaadkgacaGGPaGaeyOeI0IaamOzaiaacIcacaWGHbGaaiykaiaac2facaGGUaaaaa@5602@

Proof:  Define a function h on [a,b] by

h(x)=f(x)[g(b)−g(a)]−g(x)[f(b)−f(a)]. MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIgacaGGOaGaamiEaiaacMcacqGH9aqpcaWGMbGaaiikaiaadIhacaGGPaGaai4waiaadEgacaGGOaGaamOyaiaacMcacqGHsislcaWGNbGaaiikaiaadggacaGGPaGaaiyxaiabgkHiTiaadEgacaGGOaGaamiEaiaacMcacaGGBbGaamOzaiaacIcacaWGIbGaaiykaiabgkHiTiaadAgacaGGOaGaamyyaiaacMcacaGGDbGaaiOlaaaa@575C@

We have reduced the problem to showing that h′(c)=0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH9aqpcaaIWaaaaa@3F59@  for some c∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadogacqGHiiIZcaGGOaGaamyyaiaacYcacaWGIbGaaiykaaaa@3F2D@ .  Now, h MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIgaaaa@39D8@  is continuous on [a,b] and differentiable on (a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaacIcacaWGHbGaaiilaiaadkgacaGGPaaaaa@3CC1@ . Now

h(a)=f(a)[g(b)−g(a)]−g(a)[f(b)−f(a)]=f(a)g(b)−g(a)f(b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIgacaGGOaGaamyyaiaacMcacqGH9aqpcaWGMbGaaiikaiaadggacaGGPaGaai4waiaadEgacaGGOaGaamOyaiaacMcacqGHsislcaWGNbGaaiikaiaadggacaGGPaGaaiyxaiabgkHiTiaadEgacaGGOaGaamyyaiaacMcacaGGBbGaamOzaiaacIcacaWGIbGaaiykaiabgkHiTiaadAgacaGGOaGaamyyaiaacMcacaGGDbGaeyypa0JaamOzaiaacIcacaWGHbGaaiykaiaadEgacaGGOaGaamOyaiaacMcacqGHsislcaWGNbGaaiikaiaadggacaGGPaGaamOzaiaacIcacaWGIbGaaiykaaaa@6504@

and

h(b)=f(b)[g(b)−g(a)]−g(b)[f(b)−f(a)]=−g(a)f(b)+f(a)g(b)=h(a). MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIgacaGGOaGaamOyaiaacMcacqGH9aqpcaWGMbGaaiikaiaadkgacaGGPaGaai4waiaadEgacaGGOaGaamOyaiaacMcacqGHsislcaWGNbGaaiikaiaadggacaGGPaGaaiyxaiabgkHiTiaadEgacaGGOaGaamOyaiaacMcacaGGBbGaamOzaiaacIcacaWGIbGaaiykaiabgkHiTiaadAgacaGGOaGaamyyaiaacMcacaGGDbGaeyypa0JaeyOeI0Iaam4zaiaacIcacaWGHbGaaiykaiaadAgacaGGOaGaamOyaiaacMcacqGHRaWkcaWGMbGaaiikaiaadggacaGGPaGaam4zaiaacIcacaWGIbGaaiykaiabg2da9iaadIgacaGGOaGaamyyaiaacMcacaGGUaaaaa@6ACD@

Thus, Rolle’s Theorem guarantees a c∈(a,b) MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadogacqGHiiIZcaGGOaGaamyyaiaacYcacaWGIbGaaiykaaaa@3F2D@  so that h′(c)=0 MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadIgacqGHYaIOcaGGOaGaam4yaiaacMcacqGH9aqpcaaIWaaaaa@3F59@ , and we are done.  � MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaajugGbabaaaaaaaaapeGaeSy==7gaaa@3CD5@

Theorem 8:  Let M signify a, a + MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadggapaWaaWbaaSqabeaapeGaey4kaScaaaaa@3AFF@ , a − MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiaadggapaWaaWbaaSqabeaapeGaeyOeI0caaaaa@3B0A@ , +∞ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiabgUcaRiabg6HiLcaa@3B3E@ , or −∞ MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbhv2BYDwAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaaqaaaaaaaaaWdbiabgkHiTiabg6HiLcaa@3B49@ <