Calculus/Differentiation/Basics of Differentiation/Exercises

Find the derivative of the following functions using the limit definition of the derivative.

1. ${\displaystyle f(x)=x^{2}}$

2. ${\displaystyle f(x)=2x+2}$

3. ${\displaystyle f(x)={\frac {x^{2}}{2}}}$

4. ${\displaystyle f(x)=2x^{2}+4x+4}$

5. ${\displaystyle f(x)={\sqrt {x+2}}}$

6. ${\displaystyle f(x)={\frac {1}{x}}}$

7. ${\displaystyle f(x)={\frac {3}{x+1}}}$

8. ${\displaystyle f(x)={\frac {1}{\sqrt {x+1}}}}$

9. ${\displaystyle f(x)={\frac {x}{x+2}}}$

10. Use the definition of the derivative to prove that for any fixed real number ${\displaystyle c}$ , ${\displaystyle {\frac {d}{dx}}[c\cdot f(x)]=c\cdot {\frac {d}{dx}}[f(x)]}$

Find the derivative of the following functions:

11. ${\displaystyle f(x)=2x^{2}+4}$

12. ${\displaystyle f(x)=3{\sqrt[{3}]{x}}\,}$

13. ${\displaystyle f(x)=2x^{5}+8x^{2}+x-78}$

14. ${\displaystyle f(x)=7x^{7}+8x^{5}+x^{3}+x^{2}-x}$

15. ${\displaystyle f(x)={\frac {1}{x^{2}}}+3x^{\frac {1}{3}}}$

16. ${\displaystyle f(x)=3x^{15}+{\frac {x^{2}}{17}}+{\frac {2}{\sqrt {x}}}}$

17. ${\displaystyle f(x)={\frac {3}{x^{4}}}-{\sqrt[{4}]{x}}+x}$

18. ${\displaystyle f(x)=6x^{1/3}-x^{0.4}+{\frac {9}{x^{2}}}}$

19. ${\displaystyle f(x)={\frac {1}{\sqrt[{3}]{x}}}+{\sqrt {x}}}$

20. ${\displaystyle f(x)=(x^{4}+4x+2)(2x+3)}$

21. ${\displaystyle f(x)=(2x-1)(3x^{2}+2)}$

22. ${\displaystyle f(x)=(x^{3}-12x)(3x^{2}+2x)}$

23. ${\displaystyle f(x)=(2x^{5}-x)(3x+1)}$

24. ${\displaystyle f(x)=(5x^{2}+3)(2x+7)}$

25. ${\displaystyle f(x)=3x^{2}(5x^{2}+1)^{4}}$

26. ${\displaystyle f(x)=x^{3}(2x^{2}-x+4)^{4}}$

27. ${\displaystyle f(x)=5x^{2}(x^{3}-x+1)^{3}}$

28. ${\displaystyle f(x)=(2-x)^{6}(5+2x)^{4}}$

29. ${\displaystyle f(x)={\frac {2x+1}{x+5}}}$

30. ${\displaystyle f(x)={\frac {3x^{4}+2x+2}{3x^{2}+1}}}$

31. ${\displaystyle f(x)={\frac {x^{\frac {3}{2}}+1}{x+2}}}$

32. ${\displaystyle d(u)={\frac {u^{3}+2}{u^{3}}}}$

33. ${\displaystyle f(x)={\frac {x^{2}+x}{2x-1}}}$

34. ${\displaystyle f(x)={\frac {x+1}{2x^{2}+2x+3}}}$

35. ${\displaystyle f(x)={\frac {16x^{4}+2x^{2}}{x}}}$

36. ${\displaystyle f(x)={\frac {8x^{3}+2}{5x+5}}}$

37. ${\displaystyle f(x)={\frac {(3x-2)^{2}}{\sqrt {x}}}}$

38. ${\displaystyle f(x)={\frac {\sqrt {x}}{2x-1}}}$

39. ${\displaystyle f(x)={\frac {4x-3}{x+2}}}$

40. ${\displaystyle f(x)={\frac {4x+3}{2x-1}}}$

41. ${\displaystyle f(x)={\frac {x^{2}}{x+3}}}$

42. ${\displaystyle f(x)={\frac {x^{5}}{3-x}}}$

43. ${\displaystyle f(x)=(x+5)^{2}}$

44. ${\displaystyle g(x)=(x^{3}-2x+5)^{2}}$

45. ${\displaystyle f(x)={\sqrt {1-x^{2}}}}$

46. ${\displaystyle f(x)={\frac {(2x+4)^{3}}{4x^{3}+1}}}$

47. ${\displaystyle f(x)=(2x+1){\sqrt {2x+2}}}$

48. ${\displaystyle f(x)={\frac {2x+1}{\sqrt {2x+2}}}}$

49. ${\displaystyle f(x)={\sqrt {2x^{2}+1}}(3x^{4}+2x)^{2}}$

50. ${\displaystyle f(x)={\frac {2x+3}{(x^{4}+4x+2)^{2}}}}$

51. ${\displaystyle f(x)={\sqrt {x^{3}+1}}(x^{2}-1)}$

52. ${\displaystyle f(x)=((2x+3)^{4}+4(2x+3)+2)^{2}}$

53. ${\displaystyle f(x)={\sqrt {1+x^{2}}}}$

54. ${\displaystyle f(x)=(3x^{2}+e)e^{2x}}$

55. ${\displaystyle f(x)=e^{2x^{2}+3x}}$

56. ${\displaystyle f(x)=e^{e^{2x^{2}+1}}}$

57. ${\displaystyle f(x)=4^{x}}$

58. ${\displaystyle f(x)=2^{x-3}\cdot 3{\sqrt {x^{3}-2}}+\ln(x)}$

59. ${\displaystyle f(x)=\ln(x)-2e^{x}+{\sqrt {x}}}$

60. ${\displaystyle f(x)=\ln(\ln(x^{3}(x+1)))}$

61. ${\displaystyle f(x)=\ln(2x^{2}+3x)}$

62. ${\displaystyle f(x)=\log _{4}(x)+2\ln(x)}$

63. ${\displaystyle f(x)=3e^{x}-4\cos(x)-{\frac {\ln(x)}{4}}}$

64. ${\displaystyle f(x)=\sin(x)+\cos(x)}$

65. ${\displaystyle {\frac {d}{dx}}[(x^{3}+5)^{10}]}$

66. ${\displaystyle {\frac {d}{dx}}[x^{3}+3x]}$

67. ${\displaystyle {\frac {d}{dx}}[(x+4)(x+2)(x-3)]}$

68. ${\displaystyle {\frac {d}{dx}}[{\frac {x+1}{3x^{2}}}]}$

69. ${\displaystyle {\frac {d}{dx}}[3x^{3}]}$

70. ${\displaystyle {\frac {d}{dx}}[x^{4}\sin(x)]}$

71. ${\displaystyle {\frac {d}{dx}}[2^{x}]}$

72. ${\displaystyle {\frac {d}{dx}}[e^{x^{2}}]}$

73. ${\displaystyle {\frac {d}{dx}}[e^{2^{x}}]}$

Use implicit differentiation to find y'

74. ${\displaystyle x^{3}+y^{3}=xy}$

75. ${\displaystyle (2x+y)^{4}+3x^{2}+3y^{2}={\frac {x}{y}}+1}$

Use logarithmic differentiation to find ${\displaystyle {\frac {dy}{dx}}}$:

76. ${\displaystyle y=x({\sqrt[{4}]{1-x^{3}}})}$

77. ${\displaystyle y={\sqrt {x+1 \over 1-x}}\,}$

78. ${\displaystyle y=(2x)^{2x}}$

79. ${\displaystyle y=(x^{3}+4x)^{3x+1}}$

80. ${\displaystyle y=(6x)^{\cos(x)+1}}$

For each function, ${\displaystyle f}$ , (a) determine for what values of ${\displaystyle x}$ the tangent line to ${\displaystyle f}$ is horizontal and (b) find an equation of the tangent line to ${\displaystyle f}$ at the given point.

81. ${\displaystyle f(x)={\frac {x^{3}}{3}}+x^{2}+5,\qquad (3,23)}$

82. ${\displaystyle f(x)=x^{3}-3x+1,\qquad (1,-1)}$

83. ${\displaystyle f(x)={\frac {2x^{3}}{3}}+x^{2}-12x+6,\qquad (0,6)}$

84. ${\displaystyle f(x)=2x+{\frac {1}{\sqrt {x}}},\qquad (1,3)}$

85. ${\displaystyle f(x)=(x^{2}+1)(2-x),\qquad (2,0)}$

86. ${\displaystyle f(x)={\frac {2x^{3}}{3}}+{\frac {5x^{2}}{2}}+2x+1,\qquad \left(3,{\frac {95}{2}}\right)}$

87. Find an equation of the tangent line to the graph defined by ${\displaystyle (x-y-1)^{3}=x}$ at the point (1,-1).

88. Find an equation of the tangent line to the graph defined by ${\displaystyle e^{xy}+x^{2}=y^{2}}$ at the point (1,0).

89. What is the second derivative of ${\displaystyle 3x^{4}+3x^{2}+2x}$?

91. Let ${\displaystyle f^{\prime }(x)}$ be the derivative of ${\displaystyle f(x)}$. Prove the derivative of ${\displaystyle -f(x)}$ is ${\displaystyle -f^{\prime }(x)}$.

92. Suppose a continuous function ${\displaystyle f(x)}$ has three roots on the interval of ${\displaystyle -2\leq x\leq 2}$. If ${\displaystyle f(-2)=f(2)<0}$, then what is ONE true guarantee of ${\displaystyle f(x)}$ using

(a) the Intermediate Value Theorem;

(b) Rolle's Theorem;

(c) the Extreme Value Theorem.

93. Let ${\displaystyle g(x)=f^{-1}(x)}$, where ${\displaystyle f^{-1}(x)}$ is the inverse of ${\displaystyle f(x)}$. Let ${\displaystyle f(x)}$ be differentiable. What is ${\displaystyle g^{\prime }(x)}$? Else, why can ${\displaystyle g^{\prime }(x)}$ not be determined?

94. Let ${\displaystyle f(x)={\begin{cases}2x^{2}+4,&x<-1\\ax^{3}-2x,&x\geq -1\end{cases}}}$ where ${\displaystyle a}$ is a constant.

Find a value, if possible, for ${\displaystyle a}$ that allows each of the following to be true. If not possible, prove that it cannot be done.

(a) The function ${\displaystyle f(x)}$ is continuous but non-differentiable.

(b) The function ${\displaystyle f(x)}$ is both continuous and differentiable.