PhD candidate in Economics · Paris

Derivatives Practice: Intermediate

One hundred functions to differentiate, built out of powers, roots, logs, exponentials, products and quotients. Almost none of them need the chain rule, so this is the sheet for drilling the product and quotient rules until they are automatic.

Work each one out on paper first, then open the solution to check it. The theory is in the derivatives refresher; the other sheets are on the resources page.

  1. 1.
    \( \displaystyle f(x) = 3x^2 + 2x + 1 \)
    Solution
    \( \displaystyle f'(x) = 6 x + 2 \)
  2. 2.
    \( \displaystyle g(x) = \frac{5x^2+3x}{x} \)
    Solution
    \( \displaystyle g'(x) = 5 \)
  3. 3.
    \( \displaystyle h(x) = \ln(x^2) \)
    Solution
    \( \displaystyle h'(x) = \frac{2}{x} \)
  4. 4.
    \( \displaystyle k(x) = \frac{x^2-1}{x+1} \)
    Solution
    \( \displaystyle k'(x) = 1 \)
  5. 5.
    \( \displaystyle m(x) = \frac{1}{x^3} \)
    Solution
    \( \displaystyle m'(x) = - \frac{3}{x^{4}} \)
  6. 6.
    \( \displaystyle n(x) = \sqrt{x} + x^2 \)
    Solution
    \( \displaystyle n'(x) = 2 x + \frac{1}{2 \sqrt{x}} \)
  7. 7.
    \( \displaystyle p(x) = x^3 \cdot x^2 \)
    Solution
    \( \displaystyle p'(x) = 5 x^{4} \)
  8. 8.
    \( \displaystyle q(x) = \frac{x^2+2}{x^3-1} \)
    Solution
    \( \displaystyle q'(x) = - \frac{x^{4} + 6 x^{2} + 2 x}{\left(x^{3} - 1\right)^{2}} \)
  9. 9.
    \( \displaystyle r(x) = \frac{1}{x^2+1} \)
    Solution
    \( \displaystyle r'(x) = - \frac{2 x}{\left(x^{2} + 1\right)^{2}} \)
  10. 10.
    \( \displaystyle s(x) = x^5 - 3x^2 + \frac{1}{x} \)
    Solution
    \( \displaystyle s'(x) = 5 x^{4} - 6 x - \frac{1}{x^{2}} \)
  11. 11.
    \( \displaystyle f(x) = x^2 \ln(x) \)
    Solution
    \( \displaystyle f'(x) = 2 x \ln(x) + x \)
  12. 12.
    \( \displaystyle g(x) = \frac{\ln(x)}{x} \)
    Solution
    \( \displaystyle g'(x) = \frac{1 - \ln(x)}{x^{2}} \)
  13. 13.
    \( \displaystyle h(x) = x e^x \)
    Solution
    \( \displaystyle h'(x) = x e^{x} + e^{x} \)
  14. 14.
    \( \displaystyle k(x) = \frac{e^x}{x^2} \)
    Solution
    \( \displaystyle k'(x) = \frac{x e^{x} - 2 e^{x}}{x^{3}} \)
  15. 15.
    \( \displaystyle m(x) = x^2 + \frac{1}{\sqrt{x}} \)
    Solution
    \( \displaystyle m'(x) = 2 x - \frac{1}{2 x^{\frac{3}{2}}} \)
  16. 16.
    \( \displaystyle n(x) = \frac{x^2+1}{x^2-1} \)
    Solution
    \( \displaystyle n'(x) = - \frac{4 x}{\left(x^{2} - 1\right)^{2}} \)
  17. 17.
    \( \displaystyle p(x) = \ln(x^3) \)
    Solution
    \( \displaystyle p'(x) = \frac{3}{x} \)
  18. 18.
    \( \displaystyle q(x) = x^2 - 3x + 4 + \ln(x) \)
    Solution
    \( \displaystyle q'(x) = 2 x - 3 + \frac{1}{x} \)
  19. 19.
    \( \displaystyle r(x) = \frac{x^3+1}{x^2+1} \)
    Solution
    \( \displaystyle r'(x) = \frac{x^{4} + 3 x^{2} - 2 x}{\left(x^{2} + 1\right)^{2}} \)
  20. 20.
    \( \displaystyle s(x) = e^x - x^2 + \frac{1}{x} \)
    Solution
    \( \displaystyle s'(x) = - 2 x + e^{x} - \frac{1}{x^{2}} \)
  21. 21.
    \( \displaystyle f(x) = \frac{x^2-4x+3}{x} \)
    Solution
    \( \displaystyle f'(x) = \frac{x^{2} - 3}{x^{2}} \)
  22. 22.
    \( \displaystyle g(x) = \frac{\ln(x)}{x^2} \)
    Solution
    \( \displaystyle g'(x) = \frac{1 - 2 \ln(x)}{x^{3}} \)
  23. 23.
    \( \displaystyle h(x) = x^3 \ln(x) \)
    Solution
    \( \displaystyle h'(x) = 3 x^{2} \ln(x) + x^{2} \)
  24. 24.
    \( \displaystyle k(x) = x^3 + \frac{1}{x^3} \)
    Solution
    \( \displaystyle k'(x) = 3 x^{2} - \frac{3}{x^{4}} \)
  25. 25.
    \( \displaystyle m(x) = \frac{2x+3}{x^2} \)
    Solution
    \( \displaystyle m'(x) = - \frac{2 x + 6}{x^{3}} \)
  26. 26.
    \( \displaystyle n(x) = \frac{x}{x^2+1} \)
    Solution
    \( \displaystyle n'(x) = \frac{1 - x^{2}}{\left(x^{2} + 1\right)^{2}} \)
  27. 27.
    \( \displaystyle p(x) = x\sqrt{x} \)
    Solution
    \( \displaystyle p'(x) = \frac{3 \sqrt{x}}{2} \)
  28. 28.
    \( \displaystyle q(x) = x^4 - \frac{1}{x} \)
    Solution
    \( \displaystyle q'(x) = 4 x^{3} + \frac{1}{x^{2}} \)
  29. 29.
    \( \displaystyle r(x) = x^2 + \ln(x^2) \)
    Solution
    \( \displaystyle r'(x) = 2 x + \frac{2}{x} \)
  30. 30.
    \( \displaystyle s(x) = \frac{1+x^2}{x^3} \)
    Solution
    \( \displaystyle s'(x) = - \frac{x^{2} + 3}{x^{4}} \)
  31. 31.
    \( \displaystyle f(x) = x + \ln(x) \)
    Solution
    \( \displaystyle f'(x) = 1 + \frac{1}{x} \)
  32. 32.
    \( \displaystyle g(x) = \frac{x^2-1}{x^2+1} \)
    Solution
    \( \displaystyle g'(x) = \frac{4 x}{\left(x^{2} + 1\right)^{2}} \)
  33. 33.
    \( \displaystyle h(x) = \frac{1}{x^2} \)
    Solution
    \( \displaystyle h'(x) = - \frac{2}{x^{3}} \)
  34. 34.
    \( \displaystyle k(x) = x^2 + \frac{1}{x^2} \)
    Solution
    \( \displaystyle k'(x) = 2 x - \frac{2}{x^{3}} \)
  35. 35.
    \( \displaystyle m(x) = \frac{x}{\sqrt{x}} \)
    Solution
    \( \displaystyle m'(x) = \frac{1}{2 \sqrt{x}} \)
  36. 36.
    \( \displaystyle n(x) = \frac{x^3+2x}{x+1} \)
    Solution
    \( \displaystyle n'(x) = \frac{2 x^{3} + 3 x^{2} + 2}{\left(x + 1\right)^{2}} \)
  37. 37.
    \( \displaystyle p(x) = x^2 e^x \)
    Solution
    \( \displaystyle p'(x) = x^{2} e^{x} + 2 x e^{x} \)
  38. 38.
    \( \displaystyle q(x) = \frac{1}{x^3+1} \)
    Solution
    \( \displaystyle q'(x) = - \frac{3 x^{2}}{\left(x^{3} + 1\right)^{2}} \)
  39. 39.
    \( \displaystyle r(x) = \frac{x^2-2x+1}{x} \)
    Solution
    \( \displaystyle r'(x) = \frac{x^{2} - 1}{x^{2}} \)
  40. 40.
    \( \displaystyle s(x) = x^4 - 2x + \frac{1}{x} \)
    Solution
    \( \displaystyle s'(x) = 4 x^{3} - 2 - \frac{1}{x^{2}} \)
  41. 41.
    \( \displaystyle f(x) = \ln(x^2+1) \)
    Solution
    \( \displaystyle f'(x) = \frac{2 x}{x^{2} + 1} \)
  42. 42.
    \( \displaystyle g(x) = \frac{1}{x} + \ln(x) \)
    Solution
    \( \displaystyle g'(x) = \frac{1}{x} - \frac{1}{x^{2}} \)
  43. 43.
    \( \displaystyle h(x) = \frac{x^3-1}{x^2-1} \)
    Solution
    \( \displaystyle h'(x) = \frac{x^{2} + 2 x}{\left(x + 1\right)^{2}} \)
  44. 44.
    \( \displaystyle k(x) = x^2 + x e^x \)
    Solution
    \( \displaystyle k'(x) = x e^{x} + 2 x + e^{x} \)
  45. 45.
    \( \displaystyle m(x) = \sqrt{x} + \ln(x) \)
    Solution
    \( \displaystyle m'(x) = \frac{1}{x} + \frac{1}{2 \sqrt{x}} \)
  46. 46.
    \( \displaystyle n(x) = x e^x + 1 \)
    Solution
    \( \displaystyle n'(x) = x e^{x} + e^{x} \)
  47. 47.
    \( \displaystyle p(x) = \frac{x^2+2x+1}{x+1} \)
    Solution
    \( \displaystyle p'(x) = 1 \)
  48. 48.
    \( \displaystyle q(x) = \frac{e^x}{x} \)
    Solution
    \( \displaystyle q'(x) = \frac{x e^{x} - e^{x}}{x^{2}} \)
  49. 49.
    \( \displaystyle r(x) = x^2 + \frac{1}{x^2+1} \)
    Solution
    \( \displaystyle r'(x) = 2 x - \frac{2 x}{\left(x^{2} + 1\right)^{2}} \)
  50. 50.
    \( \displaystyle s(x) = \frac{x+1}{x-1} \)
    Solution
    \( \displaystyle s'(x) = - \frac{2}{\left(x - 1\right)^{2}} \)
  51. 51.
    \( \displaystyle f(x) = x^2 + x^3 + x^4 \)
    Solution
    \( \displaystyle f'(x) = 4 x^{3} + 3 x^{2} + 2 x \)
  52. 52.
    \( \displaystyle g(x) = \frac{x^2}{\sqrt{x}} \)
    Solution
    \( \displaystyle g'(x) = \frac{3 \sqrt{x}}{2} \)
  53. 53.
    \( \displaystyle h(x) = x - \frac{1}{x} + \ln(x) \)
    Solution
    \( \displaystyle h'(x) = 1 + \frac{1}{x} + \frac{1}{x^{2}} \)
  54. 54.
    \( \displaystyle k(x) = \frac{x^3+2}{x} \)
    Solution
    \( \displaystyle k'(x) = \frac{2 x^{3} - 2}{x^{2}} \)
  55. 55.
    \( \displaystyle m(x) = \frac{x}{e^x} \)
    Solution
    \( \displaystyle m'(x) = \frac{1 - x}{e^{x}} \)
  56. 56.
    \( \displaystyle n(x) = \ln(x) + x^2 \)
    Solution
    \( \displaystyle n'(x) = 2 x + \frac{1}{x} \)
  57. 57.
    \( \displaystyle p(x) = x^3 - x^2 + \frac{1}{x} \)
    Solution
    \( \displaystyle p'(x) = 3 x^{2} - 2 x - \frac{1}{x^{2}} \)
  58. 58.
    \( \displaystyle q(x) = x^2 + \frac{2}{x} + 1 \)
    Solution
    \( \displaystyle q'(x) = 2 x - \frac{2}{x^{2}} \)
  59. 59.
    \( \displaystyle r(x) = x^5 + x + \frac{1}{x^3} \)
    Solution
    \( \displaystyle r'(x) = 5 x^{4} + 1 - \frac{3}{x^{4}} \)
  60. 60.
    \( \displaystyle s(x) = \ln(x^4) \)
    Solution
    \( \displaystyle s'(x) = \frac{4}{x} \)
  61. 61.
    \( \displaystyle f(x) = \frac{x^4-1}{x^2+1} \)
    Solution
    \( \displaystyle f'(x) = 2 x \)
  62. 62.
    \( \displaystyle g(x) = x^2 + x^2 \ln(x) \)
    Solution
    \( \displaystyle g'(x) = 2 x \ln(x) + 3 x \)
  63. 63.
    \( \displaystyle h(x) = x \ln(x^2) \)
    Solution
    \( \displaystyle h'(x) = 2 \ln(x) + 2 \)
  64. 64.
    \( \displaystyle k(x) = \frac{1}{x^3+2} \)
    Solution
    \( \displaystyle k'(x) = - \frac{3 x^{2}}{\left(x^{3} + 2\right)^{2}} \)
  65. 65.
    \( \displaystyle m(x) = x^3 + e^x \)
    Solution
    \( \displaystyle m'(x) = 3 x^{2} + e^{x} \)
  66. 66.
    \( \displaystyle n(x) = \frac{x^2-x+1}{x^3} \)
    Solution
    \( \displaystyle n'(x) = - \frac{x^{2} - 2 x + 3}{x^{4}} \)
  67. 67.
    \( \displaystyle p(x) = \frac{1}{\sqrt{x}+1} \)
    Solution
    \( \displaystyle p'(x) = - \frac{1}{2 \sqrt{x} \left(\sqrt{x} + 1\right)^{2}} \)
  68. 68.
    \( \displaystyle q(x) = x^2 + \frac{1}{x+1} \)
    Solution
    \( \displaystyle q'(x) = 2 x - \frac{1}{\left(x + 1\right)^{2}} \)
  69. 69.
    \( \displaystyle r(x) = x \ln(x) + x \)
    Solution
    \( \displaystyle r'(x) = \ln(x) + 2 \)
  70. 70.
    \( \displaystyle s(x) = \frac{1}{x^2-1} \)
    Solution
    \( \displaystyle s'(x) = - \frac{2 x}{\left(x^{2} - 1\right)^{2}} \)
  71. 71.
    \( \displaystyle f(x) = \frac{x^2+x+1}{\sqrt{x}} \)
    Solution
    \( \displaystyle f'(x) = \frac{3 x^{2} + x - 1}{2 x^{\frac{3}{2}}} \)
  72. 72.
    \( \displaystyle g(x) = x e^x + x^2 \)
    Solution
    \( \displaystyle g'(x) = x e^{x} + 2 x + e^{x} \)
  73. 73.
    \( \displaystyle h(x) = x^2 + \frac{1}{x^3} \)
    Solution
    \( \displaystyle h'(x) = 2 x - \frac{3}{x^{4}} \)
  74. 74.
    \( \displaystyle k(x) = x^2 - \frac{1}{\sqrt{x}} \)
    Solution
    \( \displaystyle k'(x) = 2 x + \frac{1}{2 x^{\frac{3}{2}}} \)
  75. 75.
    \( \displaystyle m(x) = x^2 + \ln(x^3) \)
    Solution
    \( \displaystyle m'(x) = 2 x + \frac{3}{x} \)
  76. 76.
    \( \displaystyle n(x) = x^3 + x + \ln(x) \)
    Solution
    \( \displaystyle n'(x) = 3 x^{2} + 1 + \frac{1}{x} \)
  77. 77.
    \( \displaystyle p(x) = x^2 \cdot x^3 \)
    Solution
    \( \displaystyle p'(x) = 5 x^{4} \)
  78. 78.
    \( \displaystyle q(x) = x^5 + \frac{1}{x^2} \)
    Solution
    \( \displaystyle q'(x) = 5 x^{4} - \frac{2}{x^{3}} \)
  79. 79.
    \( \displaystyle r(x) = \frac{x^3+1}{\sqrt{x}} \)
    Solution
    \( \displaystyle r'(x) = \frac{5 x^{3} - 1}{2 x^{\frac{3}{2}}} \)
  80. 80.
    \( \displaystyle s(x) = x + \frac{1}{x^3} \)
    Solution
    \( \displaystyle s'(x) = 1 - \frac{3}{x^{4}} \)
  81. 81.
    \( \displaystyle f(x) = x^3 - \ln(x) \)
    Solution
    \( \displaystyle f'(x) = 3 x^{2} - \frac{1}{x} \)
  82. 82.
    \( \displaystyle g(x) = \frac{e^x+1}{x} \)
    Solution
    \( \displaystyle g'(x) = \frac{x e^{x} - e^{x} - 1}{x^{2}} \)
  83. 83.
    \( \displaystyle h(x) = x^2 + e^x \)
    Solution
    \( \displaystyle h'(x) = 2 x + e^{x} \)
  84. 84.
    \( \displaystyle k(x) = \frac{x^4+x^2+1}{x} \)
    Solution
    \( \displaystyle k'(x) = \frac{3 x^{4} + x^{2} - 1}{x^{2}} \)
  85. 85.
    \( \displaystyle m(x) = x + x \ln(x) \)
    Solution
    \( \displaystyle m'(x) = \ln(x) + 2 \)
  86. 86.
    \( \displaystyle n(x) = x^2 - 1 + \frac{1}{x^2} \)
    Solution
    \( \displaystyle n'(x) = 2 x - \frac{2}{x^{3}} \)
  87. 87.
    \( \displaystyle p(x) = \frac{x^2+1}{\sqrt{x^3}} \)
    Solution
    \( \displaystyle p'(x) = \frac{x^{2} - 3}{2 x^{\frac{5}{2}}} \)
  88. 88.
    \( \displaystyle q(x) = \frac{x^2-1}{x^3+1} \)
    Solution
    \( \displaystyle q'(x) = - \frac{x^{2} - 2 x}{\left(x^{2} - x + 1\right)^{2}} \)
  89. 89.
    \( \displaystyle r(x) = x^2 \cdot \ln(x^2) \)
    Solution
    \( \displaystyle r'(x) = 4 x \ln(x) + 2 x \)
  90. 90.
    \( \displaystyle s(x) = \frac{e^x-1}{x^2} \)
    Solution
    \( \displaystyle s'(x) = \frac{x e^{x} - 2 e^{x} + 2}{x^{3}} \)
  91. 91.
    \( \displaystyle f(x) = x^2 + x^3 + \ln(x) \)
    Solution
    \( \displaystyle f'(x) = 3 x^{2} + 2 x + \frac{1}{x} \)
  92. 92.
    \( \displaystyle g(x) = x^4 - \frac{1}{x^2} \)
    Solution
    \( \displaystyle g'(x) = 4 x^{3} + \frac{2}{x^{3}} \)
  93. 93.
    \( \displaystyle h(x) = x \cdot \frac{1}{x^2+1} \)
    Solution
    \( \displaystyle h'(x) = \frac{1 - x^{2}}{\left(x^{2} + 1\right)^{2}} \)
  94. 94.
    \( \displaystyle k(x) = x^3 + \frac{1}{x} + e^x \)
    Solution
    \( \displaystyle k'(x) = 3 x^{2} + e^{x} - \frac{1}{x^{2}} \)
  95. 95.
    \( \displaystyle m(x) = \frac{\ln(x)}{\sqrt{x}} \)
    Solution
    \( \displaystyle m'(x) = \frac{2 - \ln(x)}{2 x^{\frac{3}{2}}} \)
  96. 96.
    \( \displaystyle n(x) = \frac{x^2+\ln(x)}{x} \)
    Solution
    \( \displaystyle n'(x) = \frac{x^{2} - \ln(x) + 1}{x^{2}} \)
  97. 97.
    \( \displaystyle p(x) = \frac{1}{x^2+\ln(x)} \)
    Solution
    \( \displaystyle p'(x) = - \frac{2 x^{2} + 1}{x \left(x^{2} + \ln(x)\right)^{2}} \)
  98. 98.
    \( \displaystyle q(x) = x^4 + x^2 + 1 \)
    Solution
    \( \displaystyle q'(x) = 4 x^{3} + 2 x \)
  99. 99.
    \( \displaystyle r(x) = \frac{1}{\sqrt{x^2+1}} \)
    Solution
    \( \displaystyle r'(x) = - \frac{x}{\left(x^{2} + 1\right)^{\frac{3}{2}}} \)
  100. 100.
    \( \displaystyle s(x) = x^2 + \frac{1}{x^4} \)
    Solution
    \( \displaystyle s'(x) = 2 x - \frac{4}{x^{5}} \)