y = The Chain Rule Show all work. No calculator unless otherwise stated. If asked to Explain your answer, write in complete sentences.

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Mariusz Laskowski
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1 The Chain Rule Show all work. No calculator unless otherwise stated. If asked to Eplain your answer, write in complete sentences. 1. Find the derivative of the functions y 7 (b) (a) ( ) y t 1 + t 1 (c) 1 y t (d) y csc (e) y sec (π t 1) (f) y sin + sin (g) y tan _ 1 (h) r sec θ tan θ

2 . Evaluate the derivative of the function at the given point. t + (a) st ( ) t + t+ 8 at (, 4 ) (b) f ( t) t 1 at ( 0, ) (t ) 1 t + ()( t 1) ( t + )(1) s f ( ) t t + t + 8 ( t 1) s ( ) 4 f ( t) 5 ( t 1) f (0) 5. Determine the point(s) in the interval ( 0, π ) at which the graph of ( ) cos sin horizontal tangent. ( sin 1)(sin + 1) 0 f ( ) sin + cos 0 sin 1 sin 1 sin + ( cos sin ) 0 sin + (1 sin ) 0 f + has a sin + (1 sin ) 0 4 sin sin + 0 f() has horizonta tangents at: π, + 1 6, 5 π 6, 1 & π, 0 4. Find the second derivative of the function (a) ( ) ( ) f 1 (b) ( f ) sin

3 5. If h( ) tan, evaluate h ( ) π ʹ ʹ at, 6 6. If g ( 5), gʹ ( 5) 6, h ( 5), and hʹ ( 5), find ( 5) If it is not possible, state what additional information is required. g( ) (a) f ( ) (b) f ( ) g( h( ) ) h ( ) f ʹ (if possible) for each of the following. (c) f ( ) g( ) h( ) ( ) ( ) (d) f ( ) g( ) (e) f ( ) g + h( ) (f) ( ) ( ) + ( ) f g h 7. Find the derivative of f ( ) sin +cos two different ways. Simplify. What s the moral of THIS story?

4 8. Show that if f ( ) tan and g( ) sec, then fʹ ( ) gʹ ( ). Why do you think that is? 9. Show that the derivative of an odd function is an even function. That is if f ( ) f ( ), then fʹ ( ) fʹ ( ). What type of function do you think the derivative of an even function is? Can you prove it? 10. Use the fact that g( ) d fact to find 4 d. g () to prove that d g d g( ) g( gʹ ( ) ) (), ( g ) 0. Now use this

5 11. What is the largest value possible for the slope of the curve of y sin _? Justify. 1 Find the equation of the lines that are tangent and normal to the curve y tan(π /4) at 1.

6 Answers: Derivatives of Trigon ometric Functions 4 sin 5 cos sin cos sin cos sin sin cos sin sec tan sec tan sec sec sin or sec tan sin cos 1 6 sin cot 4 csc 1 cot cot csc 1 csc cot sin 1 cos 6 sin 1 cos sin 6 sec 6 tan 6

7 1 Derivatives of Ln & Eponential Functions Find the derivative. 1. y ln( ) 9. y ln(ln ) 16. y e 4. y e sin 1. y ln 10. y ln 17. y e 5. ln( ) y e. y ln( + ) y ln 18. y e e y lne y ln( cos ) 1. y ln 7( )( ) 19. y e 7. e y 5. y (ln ) 1. y ln y e 4 8. y sin( e ) y ln 14. y ln ( 5) 1. y e e 9. y e + 7. y ln( + ) 15. y ln +. y e e y e + 0. ( ) ln( y ). y e

8 B Ly0Y1FQ KiuItKa HSAoufAtUw4aur 7eO oldlkce. P rarlil1 ZrOikgpht4s4 ir7ecsiearkvge4dy.g 5 smazdjey WwOiQtlhC tiznbf5iknuidtnej kcjaplqcaurlzuqs1.i Worksheet by Kuta Software LLC Differentiation - Natural Logs and Eponentials 1) y ln d 1 ) y e d e 6 ) y ln ln 4 4) y ln ln d 1 ln ln 4 8 d 1 ln 1 ln 9 5) y cos ln 4 d sin ln sin ln 4 1 6) y e e d ee e 6e e 6 + 7) y e (4 + 5) d e(4 + 5) (4 + 5) 1 4 e (4 + 5) (4 + 5) 8) y ln 4 ( 4) d ln 4 + ( 4) ln ) y ln ( 44 5 ) d ( ( 1) ( ) ) 1 (Rules of logarithms used) 4 10) y e5 e 4 + d 4 e5 (4 + ) (0 8) 4e 54 4 (5 ) (Rules of eponents used)

9 7 yh0l1lc IKjuGtkan 5ScopfptDw6aurie6 8LSLACi.7 p 8AmlelO WrAiOgNhWtGsz rrjelsyewrav1ekdp.z v umnandger wriltphf lignsfniznoituee hcta8l7ciuklrusp.s Worksheet by Kuta Software LLC Differentiation - Logs and Eponentials 1) y 4 44 d 4 44 ln 4 16 ) y log ln 4 d 1 ln 6 ln ) y 4 5 d 4 5 ln ln ) y log 4 d 1 4 ln 8 ln 5) y log ( 5 + 5) 5 d 1 ( 5 + 5) 5 ln 5(5 + 5) ln ( 5 + 5) 6) y log 5 ( 5 ) d 1 ( 5 ) ln 5 ( 5 ) ln 5 ( 5 ) 7) y (4 + ) d (4 + ) 4 ln 4 9 (4 + ) 4 ln 4 8) y (4 + 1) d 4 ( + 1) ln ( 4 + 1) 4 4 (4 + 1) + 1 ( 4 + 1) ln cos 4 9) y d 4 cos ln 1sin cos sin 4 ln 10) y log 5 tan 4 4 d 1 tan 4 4 ln 5 sec sec 4 4 tan 4 4 ln 5

10 B Ly0Y1FQ KiuItKa HSAoufAtUw4aur 7eO oldlkce. P rarlil1 ZrOikgpht4s4 ir7ecsiearkvge4dy.g 5 smazdjey WwOiQtlhC tiznbf5iknuidtnej kcjaplqcaurlzuqs1.i Worksheet by Kuta Software LLC Kuta Software - Infinite Calculus Differentiation - Natural Logs and Eponentials Differentiate each function with respect to. 1) y ln d 1 ) y e d e 6 ) y ln ln 4 4) y ln ln d 1 ln ln 4 8 d 1 ln 1 ln 9 5) y cos ln 4 d sin ln sin ln 4 1 6) y e e d ee e 6e e 6 + 7) y e (4 + 5) d e(4 + 5) (4 + 5) 1 4 e (4 + 5) (4 + 5) 8) y ln 4 ( 4) d ln 4 + ( 4) ln ) y ln ( 44 5 ) d ( ( 1) ( ) ) 1 (Rules of logarithms used) 4 10) y e5 e 4 + d 4 e5 (4 + ) (0 8) 4e 54 4 (5 ) (Rules of eponents used) Create your own worksheets like this one with Infinite Calculus. Free trial available at KutaSoftware.com

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