Find the Derivative by Definition

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

Prove the Constant Rule

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 by Rules

Find the derivative of the following functions:

Power Rule

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

Product Rule

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

Quotient Rule

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Chain Rule

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

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

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

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

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

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

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

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

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

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

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

Exponentials

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

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

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

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

Logarithms

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

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

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

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

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

Trigonometric functions

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

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

More Differentiation

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

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

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

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

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

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

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

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

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

Implicit Differentiation

Use implicit differentiation to find y'

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

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

Logarithmic Differentiation

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

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

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

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

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

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

Equation of Tangent Line

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.

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

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

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

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

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

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

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

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

Higher Order Derivatives

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