2.5 EXERCISES 1-6 . Write the composite function in the form f(g(x)). 15. y = x sec kx 16. y = 3 cot ne [Identify the inner function u = g(x) and the outer function y = f(u).] Then find the derivative dy / dx. 17. f(x) = (2x - 3)4(x2 + x + 1)5 1. y = V1 + 4x 2. y = (2x3 + 5)4 18. g(x) = (x2 + 1)3 (x2 + 2)6 19. h(t) = (t + 1) 2/3 (2+2 - 1)3 3. y - tan TX 4. y = sin(cot x) 20. F(t) = (3t - 1)*(2t + 1)-3 5. y = v sin x 6. y = sin Vx x + 1 s + 1 21. y = x 2 - 22. f(s) = 1 52+ 4 7-42 . Find the derivative of the function. 7. F(x) = (x4+ 3x2 - 2)5 8. F(x) = (4x - x2) 100 23. y = sin(x cos x) 24. f(x) - 7 -3x 9. F(x) = 1 - 2x 10. f(x) - (1 + sec x)? (y - 1)4 25. y r2 + 1 26. G(y) = (v2 + 2y)' 11. f(z) - 72 + 1 12. f(t) = $1 + tant U 27. y = sinv1 + x2 28. F(v) = 13 + 1 13. y = cos(a' + x ) 14. y = a' + cos x Unless otherwise noted, all content on this page is @ Cengage Leaming. rave 276.5% (14/ of 650) 29. y = sin(tan 2x) 30. y = sec-(me) (a ( b 31. y = sec x + tan x 32. y = x sin X 56. Le 1 - cos 2x (a 33. y = 34. y = (ax + x2 + 62 ) -2 1 + cos 2x (b 35. y = cot (sin () 36. y = sin(sin(sin x)) 57. If 37. y = [x2 + (1 - 3x) 5]3 38. y = Vx + \\x+ vx ead (a) 39. g(x) = (2r sin rx + n)P 40. y = cost(sin x) 41. y = cos vsin(tan Trx) 42. y = [x + (x + sin x)3]4 43-46 . Find the first and second derivatives of the function. 43. y = cos(x-) 44. y = cos x 4x 45. H(t) = tan 3t 46. y = Vx + 1 58. If f 47-48 - Find an equation of the tangent line to the curve at h(x) the given point. estin 47. y = sin(sin x), (77, 0) 48. y = V1 + x3, (2, 3) (a) h 49. (a) Find an equation of the tangent line to the curve y = tan(7rx /4) at the point (1, 1). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen. 50. (a) The curve y = [x |/V2 - x? is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (1, 1). (b) Illustrate part (a) by graphing the curve and the tangent 59. If g(x line on the same screen. g'(3). 51. Find all points on the graph of the function47-48 - Find an equation of the tangent line to the curve at the given point. 47. y = sin(sin x), (7, 0) 48. y = V1 + x3, (2, 3) 49. (a) Find an equation of the tangent line to the curve y = tan(Tx /4) at the point (1, 1). (b) Illustrate part (a) by graphing the curve and the tangen line on the same screen. 50. (a) The curve y = [x|/V2 - x2 is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (1, 1). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen. 51. Find all points on the graph of the function f(x) = 2 sin x + sin x at which the tangent line is horizontal. 52. Find an equation of the tangent line to the curve y = V3 + x2 that is parallel to the line x - 2y = 1. 53. If F(x) = f(g(x)), where f(-2) = 8, f' ( -2) = 4, f'(5) = 3, g(5) = -2, and g'(5) = 6, find F' (5). 54. If h(x) = V4 + 3f(x), where f(1) = 7 and f' (1) = 4, find h' (1). 55. A table of values for f, g, f', and g' is given. X f (x) g(x) f'(x) g'(x) - - W W NI Inless thanw/so noted all content on this nano is ? Monnan76.5% SECTION 2.5 THE CHAIN RULE (a) If h(x) = f(g(x)), find h'(1). (b) If H(x) = g(f(x)), find H'(1). 56. Let f and g be the functions in Exercise 55. (a) If F(x) = f(f(x)), find F'(2). (b) If G(x) = g(g(x)), find G'(3). 57. If f and g are the functions whose graphs are sho u(x) = f(g(x)), v(x) = g(f(x)), and w(x) = glg(x each derivative, if it exists. If it does not exist, exp (a) u'(1) (b) v'(1) (c) w'(1) 9 58. If f is the function whose graph is shown, let h(x) = f(f(x)) and g(x) =f(x?). Use the graph of estimate the value of each derivative. (a) h'(2) (b) g'(2)y = f(x) gent urve rent 59. If g(x) = vf(x) , where the graph of f is shown, evaluate g'(3). f 60. Suppose f is differentiable on R and a is a real number. Let F(x) = f(x") and G(x) = [f(x)]". Find expressions for (a) F'(x) and (b) G'(x). 61. Let r(x) = f(g(h(x))), where h(1) = 2, g(2) = 3, h'(1) = 4, g'(2) = 5, and f'(3) = 6. Find r' (1). 62. If g is a twice differentiable function and f(x) = xg(x?), find f" in terms of g, g', and g". 63. Find the 50th derivative of y = cos 2x. rave N