SPRING 2015 Differentiation Practice (EXTRA PROBLEMS) 1

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1 SPRING 2015 Differentiation Practice (EXTRA PROBLEMS) 1 WARNING: These are EXTRA problems, which means you have to do all the homework, webassign and NYTI problems before doing this. Also You ll have to have the product rule, quotient rule, chain rule and all basic derivatives memorized before trying this. DO NOT TRY these problems without susbstantial preperation. These problems DO NOT CARRY any credit. 1. Find the derivative of the functions below using the power rule and your knowledge on basic derivatives. (a) f() = e e sin e (b) f() = e π + π + e sin 2 1 (c) f() = e (d) f() = e sin + e 2 2e2 5 + tan 1 () 3 25 (e) f() = 7 + csc e log Find the derivative of the functions below without using the quotient rule. (a) f() = (2 + 1) 2 (b) f() = (e + ) 2 (c) f() = (e + 1) 2 e (d) f() = (2 3)( + 1) (e) f() = ( 3 + 1)

2 3. Find the derivative using the product rule. (a) f() = 2 sin 1 9 (b) f() = 3 3 e (c) f() = 9 sin 1 cos (d) f() = e sec (e) f() = 2 e sin 1 (f) f() = 3 5 (1 + ) (g) f() = e tan 9 (sin 1 ) log 5 (h) f() = csc sin (i) f() = sin e 3 2 log 3 (j) f() = (sin )3 4. Find the derivative using the quotient rule. (a) f() = (b) f() = (c) f() = (d) f() = e e sin sin 1 (3 csc + 1) (e) f() = 2 + sec 3e sin + e (f) f() = 2 cos (g) f() = sec (h) f() = 3 cos 1 9 tan 1 (i) f() = 3 e 2 sin (j) f() = tan

3 5. Find the derivative using the product rule and the quotient rule, where ever necessary. (a) f() = e sin (b) f() = cos ln (c) f() = log 3 csc 1 (d) f() = e cos ln (1 + ) (e) f() = 3 tan 1 12 cot (f) f() = e ln + 3 (g) f() = 5 tan 1 e sin cos ln (h) f() = 12 tan 1 ( ) tan 1 (i) f() = 1 + ( ) sec (j) f() = 1 3 sin 1 e csc 6. Find the derivative using the chain rule. (a) f() = e sin (b) f() = 9 tan 1 (c) f() = 3 sin2 ) (d) f() = tan 1 ( 1 (e) f() = ln(cos 2 tan ) (f) f() = ecsc 2 1 (g) f() = 5 tan 2 (h) f() = tan 3 (sin ) (i) f() = sec 3 (e tan ) cot(csc ) (j) f() = 10 (k) f() = log 2 (sin 1 (e tan 1 (5 ) )) 3

4 7. Find the derivative using the product rule and the chain rule whenever necessary. (a) f() = 10 cot (b) f() = 9 (e sin ) (c) f() = sec(e cot2 ) (d) f() = 10 9 e tan log 3 (e) f() = ln sec(e tan 1 ) 8. Find the derivative using the chain rule and quotient rule whenever necessary. (a) f() = e sin 2 + tan 1 e esin2 (b) f() = ( 3 + 1) 9 ( e sin2 (c) f() = 2 + (d) f() = (e) f() = 9. Find the derivative. ) 3 1/3 tan 1 (sec(3 2 )) csc(e sin2 ) log 17 (tan 1 ( 2 )) (a) f() = csc2 ( tan ) tan(csc ) ( cot )/(log(sin ) (b) f() = e ( ) sin 2 (c) f() = tan 1 9 (ln )2 ( ) tan 1 ( 2 ) (d) f() = sec e csc 2 (e) f() = sec 3 ( 2 e 2 ) tan 1 (e 2 ) (f) f() = 3 sin 4

5 10. Differentiate implicitly and find dy d. (a) y 2 + 2e = 2 (b) 22 y 2ey = 9e 9y (c) ln(y) + y ln = 3 log 5 y (d) ln y + y ln 9 ln y = 0 (e) 3 y 3 y + y tan 1 = 3 (f) y y 2 y 3 y y y = 0 (g) y + tan y = 0 (h) 2 y + sin y = cos(y) (i) y sin 1 = 9y (j) sec y + sec 1 (y) 9e y ln = Differentiate and find dy d (a) y = sin log 3 (b) y = (ln ) tan 1 (c) y = (e ) tan (d) y = sin sec (e) y = csc tan 1 (f) y = tan 1 y (g) y = y (h) y = sin () (i) y = tan(sin 1 ( y )) (j) y = e tan (k) y = e tan sin (l) y = esin ( tan ) ln (sin ) (m) y = etan e tan 1 (ln ) sin 2 (n) y = 5 (tan 1 ) 3 e 9 9 e ln e e (o) y = tan 1 (ln )e 9 by logarithmic differentiation. 5

6 12. Let g() be the inverse of f() below. Find each of the values indicated. (a) f() = ( 2 + 1)e + 2 cos 1. Find g (2). (b) f() = Find g (11) (c) f() = log 3. Find g (5) (d) f() = sin 2 2 sec. Find g ( 3/2) (e) f() = 4 tan 1 () e 2. Find g (π e) 13. Find the slope of the tangent line of the following functions at the given point. (a) f() = 2 e (e ) ln 5 at = 1 (b) f() = tan 3 (csc ) at = π/4 (c) f() = at = 1 (d) f() = ln at = e3 (e) f() = 2 π π at = 1 (f) y 3 2 = 2y 1 at the point (2, 1) (g) e y ( 2 + cos )y = 3 at the point (0, 2) (h) 3 + y 2 + y + e 1 = 8 at the point (1, 3) 14. Find the value of the derivative at the indicated point of the function defined below. (a) H() = e 2 f(). It is given that f(2) = 1 4, and f (2) = 1. find H (2). (b) h() = [f() 4] 2. It is given that f(2) = 1 2 and f (2) = 1, find h (2). (c) G() = 2 f() f (). It is given that f(1) = 1, f (1) = 2 and f (1) = 3, find G (1) (d) H() = 2 f() e. It is given that f(0) = 1 and f (0) = 1. Find H (0). 15. Find the equation of the tangent line at the indicated point of the function f(). (a) f() = e3 e + 1 at = 0 (b) f() = sin at = π/4 (c) f() = csc at = π/6. (d) f() = e2 (2 + 3) 2 at = 0, use logarithmic differentiation. 3 + tan (e) f() = [g( 2 ) + 2] it is given that g(1) = 1 and g (1) = 2, find the equation of the tangent line to f() at = 1. 6

7 16. Find the equation of the normal line at the indicated point of the function f(). (a) f() = (e2 + 1) 2 at = 0 e (b) f() = 3 3 at = 2 (c) f() = e sec2 at = π/4 (d) f() = 4/3 + 2 at = Find the values of the points where the following functions have horizontal tangent lines. (a) f() = ( ) 4 ( 1) 2 (b) f() = 2 1/3 + 4/3 (c) f() = (d) f() = + cos on the interval [0, π] 2 (e) f() = (3 + 6) find the second derivative of the following functions (a) f() = (b) f() = (c) f() = tan 2 (d) f() = tan 1 (e ) (e) f() = sec( 2 ) 19. Find the value of the following limits using the definition of the derivative. (a) lim sin(π/2 + ) 1 (b) lim (c) lim 0 log (d) lim 3 ( + 3) 1 0 tan() 3 (e) lim π/3 π/3 7

8 20. Finally, as a refresher, find the derivative using the limit definition (a) f() = + 1 (b) f() = + 3 (c) f() = sin 1 (d) f() = + 2 (e) f() = ( 2 + 2) (f) f() = 1 8