Differentiation 140 Chapter 2 3rt3 See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. 69. y= 26 sec 4x, (0, 25) 70, v 2.4 Exercises Finding an Exercises 71-78, (a) find an line to the graph of the fun (b) use a graphing utility to its tangent line at the point feature of a graphing utility Equation of Finding a Derivative of a Trigonometric Function In Exercises 35-54, find the derivative of the trigonometric function. 36. y = sin X CONCEPT CHЕСК 1. Chain Rule Describe the Chain Rule for the composition of two differentiable functions in your own words. 35. y=cos 4x 38. h(x) sec 6x 40. y csc(1 - 2x)2 42. g(e) = sec(0) tan(0) 71. f(x) = /2x2 7, (4, 5) 73. y= (4x3 + 3)2, (-1, 1) 72. 37. g(x) = 5 tan 3x 2. General Power Rule What is the difference between 39. y= sin(Tx)2 41. h(x) = sin 2x cos 2x 74. the (Simple) Power Rule and the General Power Rule? 76. COS V 75. f(x) = sin 8x, (T, 0) 44. g(v) = cot x Decomposition of a Composite Function In Exercises 3-8, complete the table. 43. f(x)sin x CSC V 78. 46. g(t) = 5 cos2 t 45. y=4 sec2x 47. f(e) sin 20 49. f(t) 3 sec (nt - 1)2 77. f(x) = tan2 x, 48. h(t) = 2 cot2(t + 2) y = f(g(x)) u g(x) y = f(u) AFamous Curves In Exercises 79 50. y 5 cos(x)2 of the tangent line to the graph at t graphing utility to graph the functio point in the same viewing window. 3. y=(6x 5)4 ес A 52. y = cos(5x + csc A 4. y = 3/4x+ 3 51. y sin(3x +cos A 54. y cossin(tan mx) 1 5. y 3x+ 53. y sincot 3x A 80 В 79. Semicircle AFinding a Derivative Using Technology In Exercises 55-60, use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the 2 6. y t10 f(x) = 25 - x y 7. y csc3 x 5x 8. y sin 2 graph of the derivative. (3, 4) 4 x1 x2 1 Finding 2x a Derivative In Exercises 9-34, find 2 55. y 56. у %3 the derivative of the function. x+1 X 2 -6-4 -2 xt 1 58. g(x) = x - 1 + +1 57. y = -4 9. y= (2x- 7)3 10. y 5(2 x)4 12. f(t) (9t + 2)2/3 14. g(x) = 4-3x2 16. y 29 - x X 11. g(x) = 3(4 - 9x)5/6 COS TX +1 1 81. Horizontal Tangent Line interval (0, 2T) at which the g has a horizontal tangent. 59. у 3 60. y x2 tan 13. h(s) = 25s2 + 3 15. y 3/6x 1 Slope of a Tangent Line itExercises 61 and 62, find the slope of the tangent line to the sine function at the origin. Compare this value with the mmber of complete cycles in the interval [0, 27]. 82. Horizontal Tangent Li which the graph of 17. у 18. s(t)45t 2 6 19. g(s)= 2) f(x)= 1 20. y= (t 2)4 61. I 1 y sin 3x 21. y = 1 2 22. g(t) y=sin has a horizoal tangent. /3x +5. 23. f(x) = x(x - 2)7 25. y x1- 2-2 24. f(x)= x(2x - 5)3 26. y 16- 1 Finding a Secend Derivati second derivative of the functi 1 2т 37T 2T 2 -1 27. y 83. f(x) = 5(2 -2t 28. y 11 -2+ 1 4 - (3) 85. f(x) = 11x t 512 29. g(x) = Finding the Slope of a Graph In Exercises 63-70, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. 30. h(t) 87. f(x) = sin 1 + t 4 31. s(t) = t + 3 Evaluating a Second D evaluate the second deriva 63. y x2+8x, (1,3) 64. y 65. f(x) = 5(x3 - 2)-1, (-2,-3) 3x2 2 32. gx) =2x +3. /3x3 +4x, (2, 2) point. Use a computer algeb 33. f(x) = ((x2 + 3)5+ x2 1 66. f(x) 3x) 89. hx) 4. 16/ 34. g(x) = (2 + (x +1)4)3 (3.x + 1)3,1. 4 67. y x2)2' (0, 1) 4 68. y x22r)3 (1,-4) 91. f(x) = cos x2, (0, 1) XI2 S A AAAA AAA AA