Differentiation of Exponential Functions

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1 Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009

3 Introduction to Diffrntiation of Eponntial Functions Statmnt of Prrquisit Skills Complt all prvious TLM moduls bfor bginning this modul. Rquird Supporting Matrials Accss to th World Wid Wb. Intrnt Eplorr 5.5 or gratr. Macromdia Flash Playr. Rational Why is it important for you to larn this matrial? Larning Outcom Whn you complt this modul you will b abl to Larning Objctivs u. Find th drivativ of th ponntial function y b whr u f( ). u. Find th drivativ of y whr u f( ).. Us th chain rul whn finding th drivativ of ponntial functions. 4. Find th drivativ of product functions involving ponntials. 5. Find th drivativ of quotint functions involving ponntials. 6. Find th drivativ of ponntial functions containing trigonomtric rlations. 7. Find succssiv drivativ of ponntial functions. 8. Find th slop of an ponntial curv at a givn point. 9. Find maimum, minimum and/or inflction points for ponntial functions. Connction Activity Modul C Diffrntiation of Eponntial Functions

4 OBJECTIVE ONE Whn you complt this objctiv you will b abl to Find th drivativ of th ponntial function Eploration Activity Formula Drivation Bgin with ln ( ) Diffrntiat both sids of this quation ( ) Multiply by ( ) Now apply th chain rul to obtain d d u ( ) u du d Now for othr bass w us th fact that ln a a y u b whr u f( ) Thrfor a ln a ln a ln a ln a ln a ln a ln a ( ) ( ) ( ) ( ) ( ) Now apply th chain rul to obtain d d du i a (Formula ) d u u ( a ) a ln. a Modul C Diffrntiation of Eponntial Functions

5 EXAMPLE Find for 4 y y 0 + y ln ln0 0 EXAMPLE ( ) If y + find dy / d and valuat it at both and 0. Using formula from th prvious pag: y ln + 4 ln ( ) y 4 ( ) ln y (0) ln.7 Modul C Diffrntiation of Eponntial Functions

6 Eprintial Activity On Find th drivativ of th following functions:. y. y. y 4. y 5. y 7 ( ) 5 + ( ) ( ) 5 5 (4 ) 6 6. y 0 For th following find dy / d ; also find th valu of dy / d at th givn valu for y ; ( + + ) y 0 ; y 4 ; ( ) y 6 ;.7 4 y ; 0. ( ) y 5 ; 0. Eprintial Activity On Answrs (5 )(ln 5) ( )(ln ) + + ( 4 + )(4 )(ln 4) ( )(5 )(ln 5) ( 5 )(7 )(ln7) 6 6. (6)(0 )(ln 0) Modul C Diffrntiation of Eponntial Functions

7 OBJECTIVE TWO Whn you complt this objctiv you will b abl to u Find th drivativ of y whr u f( ). Eploration Activity If th function has th natural bas,, thn formula from objctiv can b usd. W simply rplac th bas b with and th formula bcoms; y u dy u du ( ) ( ln ) d d but notic, if you tak th natural log of bas th rsult is. Thrfor th formula for finding th drivativ of an ponntial function can b shortnd to: u if y dy u du thn ( ) (Formula ) d d Th two stps ar:. Th first part of th drivativ is th sam as th function.. Thn multiply by th drivativ of th ponnt. EXAMPLE If y 4, find dy d dy d /. (function) (drivativ of th ponnt) 4 ( )( 4) 4 4 Modul C Diffrntiation of Eponntial Functions 5

8 EXAMPLE If 4 y +, find dy / d and valuat it at 0 and at. 4+ ( )( 4) dy d dy d dy d at 0, ( )( 4) at, ( )(0) Modul C Diffrntiation of Eponntial Functions

9 Eprintial Activity Two Find dy / d for ach of th following: y y y y 7. If (7 ) 5 y + 5 y 5 6 y find dy / d and valuat th drivativ at If y find dy / d and valuat th drivativ at If y find dy / d and valuat th drivativ at If y find dy / d and valuat th drivativ at Find dy / d and valuat it at 0. 7 for:. Find dy / d and valuat it at. 9 for: y y Eprintial Activity Two Answrs.. 7 7( ) ( ). 6 ( ) 4. / ( ) 5 5. ( 4 5)( + + ) ( ) Modul C Diffrntiation of Eponntial Functions 7

10 OBJECTIVE THREE Whn you complt this objctiv you will b abl to Us th chain rul whn finding th drivativ of ponntial functions. EXAMPLE If (+ ) y, find y Us formula from objctiv and th chain rul. (+ ) y ()(+ ) () 6(+ ) (+ ) EXAMPLE ( ) If y 0, find y and valuat it at. ( ) y ln0 0 0 ( ) ( ) ( ) ( ) 6 ln0 0 Evaluating at y Modul C Diffrntiation of Eponntial Functions

11 Eprintial Activity Thr. Find dy / d for:. Find th drivativ of. Find th drivativ of ( 48) y + (9 65) 4. Givn y find y 5. Find th drivativ of ( 5 0 ) y ( ) y y ( 9 0) (6 6) 6. Givn y find dy / d and valuat it at.9. (5 0) 7. Givn y find dy / d and valuat it at.5. ( 4) 8. Givn y find dy / d and valuat it at.9. ( ) 9. Givn y + find ( ) 0. Givn y 0 find 4 dy / d and valuat it at.7. dy / d and valuat it at 0.6. Eprintial Activity Thr Answrs. ( ) ( ) ( ) ( ). 8( ) ( ) ( ) ( ) ( ) ( 9 0 ) 6. ( ) ( 6 6) ( ).9 ( 5 0) ( ) ( 4 ) ( )( ) ( ) ( ) 4 ln ( ) 0.6 Modul C Diffrntiation of Eponntial Functions 9

12 OBJECTIVE FOUR Whn you complt this objctiv you will b abl to Find th drivativ of product functions involving ponntials. Eploration Activity For som problms it will b ncssary to apply th product rul of drivativs. Th product rul stats: If y u v whr u and v ar functions of thn: dy dv du u + v d d d A rviw of this concpt may b found in modul C Drivativs by Formula EXAMPLE cos If y, find dy / d. Us formula from objctiv and apply th product rul. u and v cos dy + d cos ( sin + ) cos cos ( )( )( sin ) ( )( ) EXAMPLE If y, find dy / d and valuat it at. 5. dy + d 6 ( ) ( )( )( ) ( )(6 ) and at.5 w gt Modul C Diffrntiation of Eponntial Functions

13 Eprintial Activity Four dy / d and valuat it at 04.. ( ). Givn y (7 8) find. Find th drivativ of th following function and valuat it at 06. : ( 0. 4) y 7. Find th drivativ of th following function and valuat it at 0. : (5+ 0. ) y (6 + 7) 4. Find th drivativ of th following function and valuat it at 06. : (+ 0. 8) y 5 5. Find th drivativ of th following function and valuat it at 05. : (4 0. ) y ( + 7) y (8 6) / d and valuat it at 0.. (5 0. 4) 6. Givn find dy Eprintial Activity Four Answrs. ( ( )) ( 4 + y 0.8) ( ) ( y 0.4) 0.6 ( ) ( 5 0.) y ( ) 4. ( ) ( + y 0.8) ( ( )) ( y ) ( ) ( ) ( 5 0.4) y Modul C Diffrntiation of Eponntial Functions

14 OBJECTIVE FIVE Whn you complt this objctiv you will b abl to Find th drivativ of quotint functions involving ponntials. Eploration Activity Somtims it is ncssary to apply th quotint rul of drivativs. Th quotint rul stats: u If y whr u and v ar functions of thn v du dv v u dy d d d v A rviw of this concpt may b found in modul C Drivativs by Formula EXAMPLE 4 If y, find dy / d. 4 Us formula from objctiv and apply th quotint rul. u and v 4 4 dy 8 d 4 4 ( ) 5 ( ) ( )( )(4) ( )(4 ) Modul C Diffrntiation of Eponntial Functions

15 EXAMPLE If y, find dy d and valuat it at y and v dy d ( ) ( )( ) ( )( ) ( ) dy ( ) d and at 0.5, 05. dy 05. ( 05) d. 78 It is asir to avoid th quotint rul as follows: y ( ) y Modul C Diffrntiation of Eponntial Functions

16 Eprintial Activity Fiv. Find dy / d and valuat dy / d at 0. 7 for: y (5 8) 6. Find dy / d and valuat dy / d at 0. for: y ( ). Find dy / d and valuat dy / d at 0. 5 for: 4. Find dy / d and valuat dy / d at 0. 4 for: y 5 (4+ 7) (8+ ) y Find th drivativ of 6. Find th drivativ of y y 6 4 thn valuat it at thn valuat it at 05.. Eprintial Activity Fiv Answrs ( ) (5 ) y y 6 y (6 ) (4 ) y (5+ ) 5 (40+ ) y (5+ ) (5+ ) 4 6 y y ( + ) y + 4 y ( + ) Modul C Diffrntiation of Eponntial Functions

17 OBJECTIVE SIX Whn you complt this objctiv you will b abl to Find th drivativ of ponntial functions containing trigonomtric rlations. Eploration Activity You will nd to rfr to th ruls for finding th drivativs of th trigonomtric functions. EXAMPLE If y /. tan, find dy d Us formula from objctiv and rmmbr th drivativ of tan is sc. dy d EXAMPLE If tan ( )(sc ) y 0 sin, find dy / d and valuat at π / 6. Us formula from objctiv, and rmmbr you must find th drivativs of sin dy d π dy, 0 6 d sin (0 )(cos )(ln 0) Modul C Diffrntiation of Eponntial Functions 5

18 Eprintial Activity Si sin 7. If y find dy / d and valuat th drivativ at If y sin 7 find dy d / and valuat th drivativ at 0... If y 9 arctan 9 find dy / d and valuat th drivativ at If y cos (8 ) find dy / d and valuat th drivativ at sin 5. If y find dy / d and valuat th drivativ at Eprintial Activity Si Answrs. y 7cos7 sin y sin 7+ 7cos7.58 sin 7. ( ) 0.. y arctan cos8 8sin ( ) y y 8sin 9 ( 9sin ) y 8cos Modul C Diffrntiation of Eponntial Functions

19 OBJECTIVE SEVEN Whn you complt this objctiv you will b abl to Find succssiv drivativs of ponntial functions. Eploration Activity If can b usful in crtain applications to find th scond and third drivativs of ponntials. It is somtims asir to find th scond drivativ to tst maimum and minimum points. Also th tst of an inflction point can com from th third drivativ of a function. EXAMPLE If y 0, find th scond drivativ. dy d ( )( 0) d y 0 and 0 ( 0) d EXAMPLE 00 0 If y, find th scond drivativ and valuat it at 5.. dy d ( ) Now to find th scond drivativ w must apply th product rul: d y d ( )( )( ) ( ) + Modul C Diffrntiation of Eponntial Functions 7

20 Eprintial Activity Svn. Find th scond drivativ of th following function and valuat it at t 0.7: t y ( 8). Find th scond drivativ of th following function and valuat it at 0.67: y sin 7. Find th scond drivativ of th following function and valuat it at 0.6: y arctan 4. Find th scond drivativ of th following function and valuat it at 0.5: 4 y ln 5. Find th scond drivativ of th following function and valuat it at 0.5: 4 y 5 6. Find th scond drivativ of th following function and valuat it at 0.67: y cos Eprintial Activity Svn Answrs. t t ( ) ( ) ( ) y 8 t t t t y t (sin 7 + 7cos7 ) y ( 40sin 7 + 4cos7 ) y y ( + ) ( + ) ( + ) y 0.5 y 4 ln+ 4 4 y ln y (5 + 0 ) 4 y + 4 (40 80 ) y (cos sin ) y 8 sin Modul C Diffrntiation of Eponntial Functions

21 OBJECTIVE EIGHT Whn you complt this objctiv you will b abl to Find th slop of an ponntial curv at a givn point. Eploration Activity As statd arlir, th rsult of finding th drivativ of a function rprsnts th formula for finding th slop of a curv at a givn point. Thrfor, find th drivativs of any function, thn substitut a particular -valu and you will find th slop of th function at that point. Modul will provid you with a rviw of this concpt. EXAMPLE If y, find th slop of th curv at. dy d at, dy d. ()()( ) at, substituting into th original quation, y.78. Thrfor at th point (,.78), th slop is: Modul C Diffrntiation of Eponntial Functions 9

22 EXAMPLE If y, find th slop if. Also, dos th curv hav a horizontal tangnt, if so, find th point. What is th slop at that point? dy 6 d ( ) 6 ( ) 4 at, 6 dy () d 6. Thrfor at th point (,50.4) th slop is Now, for a horizontal tangnt th slop must b zro. Thrfor st th drivativ qual to zro and solv for. 4 ( ) 0 multiply both sids by 4 ( )( ) 0 st both factors to zro 0 0or 0or Th factor is always positiv, thrfor it can nvr b qual to zro. So is th only accptabl valu. At, y 0.09 rprsnts th point whr th curv has a horizontal tangnt and whr th slop is zro. 0 0 Modul C Diffrntiation of Eponntial Functions

23 Eprintial Activity Eight cos( 6). If y find th slop at. 0. If. If 4. If 5. If ( ) y find th slop at sin 7 y find th slop at ( 9) y find th slop at 00. y. 6 find th slop at 05 Eprintial Activity Eight Answrs cos( 6) y sin ( ) y y + y ( ) sin7 ( ) sin7 + 9 y (7cos7 ) y y ( + ) y y Modul C Diffrntiation of Eponntial Functions

24 OBJECTIVE NINE Whn you complt this objctiv you will b abl to Find maimum, minimum and/or inflction points for ponntial functions. Eploration Activity Maima or Minima To find maimum or minimum points w tak th following stps:. Find th point at which m 0 (slop 0).. Tsts: (a) Nar point tst i. If th slop changs from ( ) to (+), th point is a minimum. ii. If th slop changs from (+) to ( ), th point is a maimum. iii. If th slop dos not chang in sign th point is nithr a maimum nor a minimum. (b) Scond drivativ tst i. If y ( ), th point is a maimum ii. If y (+), th point is a minimum iii. If y 0, tst fails, us nar point tst If you nd to rviw ths tsts, rfr to modul C5 Applications of Drivativs. EXAMPLE Find any maimum and/or minimum points for: y dy ( )( )() + d ( ) Stting th drivativ to zro and solving for, ( )( + ) 0 St ach factor to zro: / Sinc is always positiv, can nvr b qual to zro. Thrfor only th point / rprsnts a possibl maimum or minimum point. Modul C Diffrntiation of Eponntial Functions

25 Tst th point /. Finding th scond drivativ of this function may b difficult, thrfor us th nar point tst. at, dy ( ) d at 0, dy 0 ( + ) d Th slop changs from ngativ to positiv. Thrfor th point ( 0., ) is a minimum point. Inflction Points From A to B th curv is concav down. A B C From B to C th curv is concav up. Th point B rprsnts th inflction point bcaus it is th point whr th curv changs its concavity. To find an inflction point:. Find th scond drivativ and quat it to zro.. Tst th point: (a) Nar point tst. i. If y changs in passing through th point, this point is thn an inflction point ii. If y dos not chang, th point is not an inflction point (b) Third drivativ tst i. If th valu of th third drivativ at th point is positiv or ngativ, thn th point is an inflction point ii. If th valu of th third drivativ is zro, th tst fails, thn us th nar point tst EXAMPLE Find any maimum, minimum and/or inflction points for th following curv: y y dy ( )( )( ) + d + ( )( ) Modul C Diffrntiation of Eponntial Functions

26 To find th maimum or minimum points, st th drivativ to zro. Thrfor: ( )( ) 0 now stting ach factor to zro: 0 and ( ) 0 0 Sinc cannot b qual to zro, w hav critical valus of 0 and. It is lft to th studnt to tst ths valus to dtrmin whthr thy ar maimum or minimum. Us th nar point tst. To dtrmin whthr th givn function has any inflction points, find th scond drivativ and st it to zro. y y ( ) y ) 0 ( ) ( solving th scond drivativ quals zro 4+ 0 using th quadratic formula ± Again it will b lft to th studnt to tst ths valus. + and Sinc finding th third drivativ of th function may not b asy, it is suggstd that th nar point tst b usd..44 and ar th coordinats of th points of inflction. Th following sktch is typical of a class of curvs which corrspond to th typ of prssion in this ampl. y Ma P.I P.I Ma Obsrv hr w hav points of inflction and ma/min points 4 4 Modul C Diffrntiation of Eponntial Functions

27 Eprintial Activity Nin (7 8). Givn y 4 find any maimum, minimum and/or inflction points.. Givn. Givn 4. Givn 5. Givn y y y 6 find any maimum, minimum and/or inflction points. 5 find any maimum, minimum and/or inflction points. find any maimum, minimum and/or inflction points. y find any maimum, minimum and/or inflction points. Eprintial Activity Nin Answrs 5. (. 0 49, ) min 5 ( , ) inflction point. (, ) ma (0, 0) inflction point (0. 46, 0. 75) inflction point (. 5774, ) inflction point. (, 6. 7) min (. 4 7, ) inflction point (. 679,. 868) inflction point (0, 0) inflction point 4. (0, 0) min (, ) (0. 95, 0. 0) (. 8, ma ) inflction point inflction point 5. (0, 0) min (,. 087) ma ( , 0. 80) inflction point (. 44, ) inflction point Modul C Diffrntiation of Eponntial Functions 5

28 Practical Application Activity Complt th Diffrntiation of Eponntial Functions assignmnt in TLM. Summary Diffrntiation of ponntial functions was covrd in dtail. Functions of th form u u y b and in particular y wr diffrntiatd alon and in combination with othr prssions. This modul prpars th studnt wll, for applid aras spcially in th fild of lctronics. 6 6 Modul C Diffrntiation of Eponntial Functions

Calculus concepts derivatives

Calculus concepts derivatives All rasonabl fforts hav bn mad to mak sur th nots ar accurat. Th author cannot b hld rsponsibl for any damags arising from th us of ths nots in any fashion. Calculus concpts drivativs Concpts involving