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Chapter 15 Additional Exercises

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					                                 Chapter 15, Additional Exercises

                         (Note: Many of the derivatives are not simplified)

For the following functions, find f’(x):

   1. f(x) = -25

Answer: f’(x) = 0

   2. f(x) = 10

Answer: f’(x) = 0

   3. f(x) = x0

Answer: f’(x) = 0

   4. f(x) = x6

Answer: f’(x) = 6x5

   5. f(x) = - x3

Answer: f’(x) = -3x2

   6. f(x) = 1/x3

Answer: f’(x) = -3/x4

   7. f(x) = 5/x

Answer: f’(x) = -5/x2

   8. f(x) = 20/x4

Answer: f’(x) = -80/x5

   9. f(x) = 1/2x6

Answer: f’(x) = -3/x7

   10. f(x) = 2/3x3

Answer: f’(x) = -2/x4
   11. f(x) = 3/4x4

Answer: f’(x) = -3/x5

   12. f(x) = -3/2x4

Answer: f’(x) = 6/x5

   13. f(x) = x7/7

Answer: f’(x) = x6

   14. f(x) = 2x5/5

Answer: f’(x) = 2x4

   15. f(x) = -5x3/3

Answer: f’(x) = - 5x2

   16. f(x) = 5x8/8

Answer: f’(x) = 5x7

   17. f(x) = 3x10/5

Answer: f’(x) = 6x9

   18. f(x) = 5x8/4

Answer: f’(x) = 10x7

   19. f(x) = -3x4 - 5x2 + 10x - 100

Answer: f’(x) = -12x3 - 10x + 10

   20. f(x) = 25x3 - 5x2 + 10x - 50

Answer: f’(x) = 75x2 - 10x + 10

   21. f(x) = x5/5 - x3/3 + x2/2 - 5

Answer: f’(x) = x4 - x2 + x

   22. f(x) = 4x3/3 - 5x2/2 + 5x/2 - 25

Answer: f’(x) = 4x2 - 5x + 5/2
   23. f(x) = 9x4/4 + 5x3/3 - 3x2/2 + 5x/4 - 100

Answer: f’(x) = 9x3       + 5x2 - 3x + 5/4

   24. f(x) = 7x5/5 - 4x3/3        - x/2

Answer: f’(x) = 7x4 - 4x2 - ½

   25. f(x) = 2.5x3 + .5x2 - x/5 + 8

Answer: f’(x) = 7.5x2 + x - 1/5

   26. f(x) = .5x6 - x5/5 + x4/8

Answer: f’(x) = 3x5 - x4 + x3/2

   27. f(x) = 30e5x -1

Answer: f’(x) = 150e5x-1

   28. f(x) = e100 – 8x

Answer: f’(x) = -8e100-8x

   29. f(x) = ex2- 5

Answer: f’(x) = 2x ex2- 5

   30. f(x) = 10,000e.05x

Answer: f’(x) = 500e.05x

   31. f(x) = 200,000e-.025x

Answer: f’(x) = -5000e-.025x

   32. f(x) = 10e- x

Answer: f’(x) = -10e-x

   33. f(x) = ex3-5x

Answer: f’(x) = (3x2 – 5) ex3-5x
   34. f(x) = ex4-5x2

Answer: f’(x) = (4x3 - 10x) ex4-5x2

   35. f(x) = ex3-4x2+ 10x

Answer: f’(x) = (3x2 – 8x + 10) ex3-4x2+ 10x

   36. f(x) = ln (25x4 – 100)

Answer: f’(x) = 100x3 / (25x4 – 100)

   37. f(x) = ln (10 – 5x3)

Answer: f’(x) = -15x2 / (10 – 5x3)

   38. f(x) = 10 ln(5x)

Answer: f’(x) = 10 (5/5x) = 10/x

   39. f(x) = 25 ln (x4 – 20x)

Answer: f’(x) = 25 *(4x3 – 20)/(x4 – 20x)]

   40. f(x) = ln (100)

Answer: f’(x) = 0

   41. f(x) = ln (5x2 – 8x + 50)

Answer: f’(x) = (10x – 8)/(5x2 – 8x + 50)

   42. f(x) = 10,000 ln (50)

Answer: f’(x) = 0

   43. f(x) = 15 ln (x)

Answer: f’(x) = 15 / x

   44. f(x) = x3 ln (x)

Answer: f’(x) = 3x2 ln(x) + (1/x) x3
   45. f(x) = ex ln(x)

Answer: f’(x) = ex ln(x) + (1/x) ex

   46. f(x) = (x5 - 1)(6x5 – x3 + 15)

Answer: f’(x) = 5x4 (6x5 – x3 + 15) + (30x4 – 3x2)(x5 – 1)

   47. f(x) = e10x – 1 ln(8x3 – 20)

Answer: f’(x) = 10e10x -1 ln(8x3 – 20) + (24x2 / 8x3 -20) e10x-1

   48. f(x) = (125 – x8) ln (25x3 – 10x2 + x)

Answer: f’(x) = (-8x7) ln (25x3 – 10x2 + x) + [(75x2 – 20x + 1) / (25x3 – 10x2 + x)] (125 – x8)

   49. f(x) = 250e-.1x ln(x5 – 8x3)

Answer: f’(x) = - 25e-.1x ln (x5 – 8x3) + [(5x4 – 24x2)/(x5 – 8x3)][250e-.1x]

   50. f(x) = 75x4 e8x – 5

Answer: f’(x) = 300x3 e8x-5 + 8e8x-5 (75x4)

   51. f(x) = 10e5 ln(100)

Answer: f’(x) = 10 (0)e5 ln(100) + (0/100)(10e5) = 0

   52. f(x) = (x5 - 8x4 + 5x2 - 1) (125 – x + 50x3)

Answer: f’(x) = (5x4 – 32x3 + 10x)(125 – x + 50x3) + (-1 +150x2)(x5 – 8x4 + 5x2 -1)

   53. f(x) = 100/(x3 – 10)

Answer: f’(x) = [(x3 – 10)(0) – 100(3x2)] / (x3 – 10)2

   54. f(x) = x / (5x3 – 50)

Answer: f’(x) = *(5x3 – 50)(1) - x (15x2)] / (5x3 - 50)2

   55. f(x) = (x8 + 10x2) / (5x3 - 8x2 + x - 1)

Answer: f’(x) = *(5x3 - 8x2 + x - 1)(8x7 + 20x) – (x8 + 10x2)(15x2 – 16x)] /(5x3 - 8x2 + x - 1)2

   56. f(x) = ex / ln(x)

Answer: f’(x) = *ln(x)ex - ex (1/x)] / [ln(x)]2
    57. f(x) = (25x2 - 3x + 30) / e10x – 5

Answer: f’(x) = *e10x – 5 (50x -3) - (25x2 – 3x + 30)(10e10x-5)] / (e10x-5] 2

    58. f(x) = 25 / (80x5 – 100)

Answer: f’(x) = *(80x5 - 100)(0) - 25(400x4)] / (80x5 – 100) 2

    59. f(x) = ln (x5 + 25x2) / 8e.5x + 10

Answer: f’(x) = [8e.5x + 10 (5x4 + 50x)/(x5 + 25x2) – ln(x5 + 25x2) (4e.5x + 10)] / (8e.5x+10) 2

    60. f(x) = 50e10-.3x / (200 – x10)

Answer: f’(x) = *(200 – x10) (-15e10-.3x) – 50e10-.3x (-10x9)] / (200 – x10) 2

    61. f(x) = (x6 - 4x3 + 25) / (.5x8 - x3 /3)

Answer: f’(x) = *(.5x8 - x3/3) (6x5 – 12x2) –(x6 – 4x3 + 25)(4x7 – x2)] / (.5x8 - x3/3) 2

    62. f(x) = e5 / ln(5)

Answer: f’(x) = *ln(5)(0e5) – e5 (0/5)] / [ln(5)] 2 = 0

    63. f(x) = (x2 – 10x + 5) 4

Answer: f’(x) = 4(x2 – 10x + 5) 3 (2x – 10)

    64. f(x) = (250 – 20x3) 10

Answer: f’(x) = 10(250 – 20x3) 9 (-60x2)

    65. f(x) = (x3 – 6x2 - x + 40) 6

Answer: f’(x) = 6(x3 – 6x2 – x + 40) 5 (3x2 – 12x - 1)

    66. f(x) = (80x5 – 25x3 + 50x) 4

Answer: f’(x) = 4(80x5 – 25x3 + 50x) 3 (400x4 -75x2 + 50)

    67. f(x) = [(x3/3) - (5x2/2) - x] 3

Answer: f’(x) = 3 *(x3/3) – (5x2/2) – x] 2 (x2 – 5x -1)

    68. f(x) = (e6x – 1 + 10x3) 5

Answer: f’(x) = 5(e6x – 1 + 10x3)4 (6e6x-1 + 30x2)
    69. f(x) = [ ln(x+1) - 5x3] 4

Answer: f’(x) = 4 * ln(x+1) - 5x3] 3 [(1/x+1) – 15x2]

    70. f(x) = 1 / (9x3 – x2 + 8) 3

Answer: f’(x) = -3 (9x3 – x2 + 8) -4 (27x2 – 2x)

    71. f(x) = ( x5 - 8x2) ½

Answer: f’(x) = (1/2)(x5 – 8x2) -1/2 (5x4 – 16x)

    72. f(x) = (25 – x6) 2/3

Answer: f’(x) = (2/3) (25 – x6) -1/3 (-6x5)

    73. f(x) = 1 / (x2 – x + 5) ½

Answer: f’(x) = (-1/2)(x2 – x + 5) -3/2 (2x – 1)

    74. f(x) = [ 1/(x+1)] 4

Answer: f’(x) = 4* 1/(x+1)] 3 [(x+1)(0) – 1(1)] / (x+1) 2

    75. f(x) = [ ex / (10-x2)] 8

Answer: f’(x) = 8* ex / (10-x2)] 7 [(10-x2)ex – ex (-2x)] / (10 – x2) 2

    76. f(x) = [(25 – x3) e5x – 1 ] 3

Answer: f’(x) = 3*(25 – x3) e5x – 1 ] 2 [(-3x2)e5x-1 + 5e5x-1(25-x3)]

    77. f(x) = [(x4 – 10) ln(x2 -1)] 5

Answer: f’(x) = 5 [(x4 – 10) ln(x2 -1)] 4 [(4x3) ln(x2-1) + [(2x)/(x2-1)](x4 -10)]

				
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