Calculus/Differentiation/Basics of Differentiation/Exercises

Find the Derivative by Definition

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

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

2. $f(x)=2x+2$

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

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

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

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

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

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

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

Prove the Constant Rule

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

Find the Derivative by Rules

Find the derivative of the following functions:

Power Rule

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

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

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

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

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

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

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

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

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

Product Rule

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

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

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

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

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

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

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

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

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

Quotient Rule

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Chain Rule

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

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

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

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

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

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

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

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

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

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

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

Exponentials

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

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

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

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

Logarithms

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

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

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

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

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

Trigonometric functions

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

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

More Differentiation

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

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

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

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

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

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

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

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

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

Implicit Differentiation

Use implicit differentiation to find y'

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

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

Logarithmic Differentiation

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

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

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

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

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

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

Equation of Tangent Line

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

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

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

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

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

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

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

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

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

Higher Order Derivatives

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

90. Use induction to prove that the (n+1)th derivative of a n-th order polynomial is 0.

Advanced Understanding of Derivatives

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

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

(a) the Intermediate Value Theorem;

(b) Rolle's Theorem;

(c) the Extreme Value Theorem.

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

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

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

(a) The function $f(x)$ is continuous but non-differentiable.

(b) The function $f(x)$ is both continuous and differentiable.