Calculus/Differentiation/Basics of Differentiation/Exercises

 ← Some Important Theorems Calculus L'Hôpital's rule → 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. $f(x) = x^2 \,$

$2x$

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

$2$

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

$x$

4. $f(x) = 2x^2 + 4x + 4 \,$

$4x+4$

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

$\frac{1}{2\sqrt{x+2}}$

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

$-\frac{1}{x^2}$

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

$\frac{-3}{(x+1)^2}$

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

$\frac{-1}{2(x+1)^{3/2}}$

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

$\frac{2}{(x+2)^2}$

Solutions

Prove the Constant Rule

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

\begin{align}\frac{d}{dx}\left[cf(x)\right] &=\lim_{\Delta x \to 0}\frac{cf\left(x+\Delta x \right)-cf\left(x\right)}{\Delta x}\\ &=c\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}\\ &=c\frac{d}{dx}\left[f(x)\right]\end{align}

Solutions

Find the Derivative by Rules

Find the derivative of the following functions:

Power Rule

11. $f(x) = 2x^2 + 4\,$

$f'(x)=4x$

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

$f'(x)=\frac{1}{\sqrt[3]{x^2}}$

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

$f'(x)=10x^4+16x+1$

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

$f'(x)=49x^6+40x^4+3x^2+2x-1$

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

$f'(x)=\frac{-2}{x^3}+\frac{1}{\sqrt[3]{x^2}}$

16. $f(x) = 3x^{15} + \frac{1}{17}x^2 +\frac{2}{\sqrt{x}} \,$

$f'(x)=45x^{14}+\frac{2}{17}x-\frac{1}{x\sqrt{x}}$

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

$f'(x)=\frac{-12}{x^5}-\frac{1}{4\sqrt[4]{x^3}}+1$

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

$f'(x)=\frac{2}{\sqrt[3]{x^2}}-\frac{0.4}{x^{0.6}}-\frac{18}{x^3}$

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

$f'(x)=\frac{-1}{3x\sqrt[3]{x}}+\frac{1}{2\sqrt{x}}$

Solutions

Product Rule

20. $f(x) = (x^4+4x+2)(2x+3) \,$

$10x^4+12x^3+16x+16$

21. $f(x) = (2x-1)(3x^2+2) \,$

$18x^2-6x+4$

22. $f(x) = (x^3-12x)(3x^2+2x) \,$

$15x^4+8x^3-108x^2-48x$

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

$36x^5+10x^4-6x-1$

$f(x) = (5x^2+3)(2x+7)\,$

$30x^2+70x+6$

$f(x) = 3x^2(5x^2+1)^4 \,$

$6x(25x^2+1)(5x^2+1)^3$

$f(x) = x^3(2x^2-x+4)^4 \,$

$x^2(2x^2-x+4)^3(22x^2-7x+12)$

$f(x) = 5x^2(x^3-x+1)^3 \,$

$5x(x^3-x+1)^2(11x^3-2x+1)$

$f(x) = (2-x)^6(5+2x)^4 \,$

$2(x-2)^5(2x+5)^3(10x+7)$

Solutions

Quotient Rule

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

$f'(x)=\frac{9}{(x+5)^2}$

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

$f'(x)=\frac{18x^5+12x^3-6x^2-12x+2}{(3x^2+1)^2}$

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

$f'(x)=\frac{x\sqrt{x}+6\sqrt{x}-2}{2(x+2)^2}$

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

$d'(u)=-\frac{6}{u^4}$

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

$f'(x)=\frac{2x^2-2x-1}{(2x-1)^2}$

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

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

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

$f'(x)=48x^2+2$

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

$f'(x)=\frac{2(8x^3+12x^2-1)}{5(x+1)^2}$

$f(x) = \frac{(3x-2)^2}{x^{1/2}} \,$

$f'(x)=\frac{(3x-2)(9x+2)}{2x^{3/2}}$

$f(x) = \frac{ x^{1/2}}{2x-1} \,$

$f'(x)=\frac{-(2x+1)}{2x^{1/2} (2x-1)^2}$

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

$f'(x)=\frac{11}{(x+2)^2}$

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

$f'(x)=\frac{-10}{(2x-1)^2}$

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

$f'(x)=\frac{x(x+6)}{(x+3)^2)}$

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

$f'(x)=\frac{x^4(-4x+15)}{(3-x)^2}$

Solutions

Chain Rule

31. $f(x) = (x+5)^2 \,$

$f'(x)=2(x+5)$

32. $g(x) = (x^3 - 2x + 5)^2 \,$

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

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

$f'(x)=-\frac{x}{\sqrt{1-x^{2}}}$

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

$f'(x)=\frac{6(4x^{3}+1)(2x+4)^{2}-(2x+4)^{3}(12x^{2})}{(4x^{3}+1)^{2}}$

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

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

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

$f'(x)=\frac{2x+3}{(2x+2)^{3/2}}$

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

$f'(x)=\frac{2x(3x^{4}+2x)^{2}}{\sqrt{2x^{2}+1}}+\sqrt{2x^{2}+1}(2)(3x^{4}+2x)(12x^{3}+2)$

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

$f'(x)=\frac{2(x^{4}+4x+2)^{2}-2(2x+3)(x^{4}+4x+2)(4x^{3}+4)}{(x^{4}+4x+2)^{4}}$

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

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

40. $f(x) = ((2x+3)^4 + 4(2x+3) +2)^2 \,$

$f'(x)=2((2x+3)^{4}+4(2x+3)+2)(8(2x+3)^{3}+8)$

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

$f'(x)=\frac{x}{\sqrt{1+x^{2}}}$

Solutions

Exponentials

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

$f'(x)=6xe^{2x}+2e^{2x}(3x^{2}+e)$

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

$f'(x)=(4x+3)e^{2x^{2}+3x}$

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

$f'(x)=4xe^{2x^{2}+1+e^{2x^{2}+1}}$

45. $f(x) = 4^x\,$

$f'(x)=\ln(4)4^{x}$

Solutions

Logarithms

46. $f(x) = 2^{x-3}\cdot3\sqrt{x^3-2}+\ln x\,$

$f'(x)=3\ln(2)2^{x-3}\sqrt{x^{3}-2}+\frac{9x^{2}2^{x-4}}{\sqrt{x^{3}-2}}+\frac{1}{x}$

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

$f'(x)=\frac{1}{x}-2e^{x}+\frac{1}{2\sqrt{x}}$

48. $f(x) = \ln(\ln(x^3(x+1))) \,$

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

49. $f(x) = \ln(2x^2+3x)\,$

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

50. $f(x) = \log_4 x + 2\ln x\,$

$f'(x)=\frac{1}{x\ln4}+\frac{2}{x}$

Solutions

Trigonometric functions

51. $f(x) = 3e^x-4\cos (x) - \frac{1}{4}\ln x\,$

$f'(x)=3e^{x}+4\sin(x)-\frac{1}{4x}$

52. $f(x) = \sin(x)+\cos(x)\,$

$f'(x)=\cos(x)-\sin(x)$

Solutions

More Differentiation

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

$30x^{2}(x^{3}+5)^{9}$

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

$3x^{2}+3$

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

$(x+2)(x-3)+(x+4)(x-3)+(x+4)(x+2)$

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

$-\frac{x+2}{3x^{3}}$

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

$9x^{2}$

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

$4x^{3}\sin x+x^{4}\cos x$

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

$\ln(2)2^{x}$

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

$2xe^{x^{2}}$

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

$2*e^{2^{x}}$

Solutions

Implicit Differentiation

Use implicit differentiation to find y'

62. $x^3 + y^3 = xy \,$

$y'=\frac{y-3x^{2}}{3y^{2}-x}$

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

$y'=\frac{y-8y^{2}(2x+y)^{3}-6xy^{2}}{4y^{2}(2x+y)^{3}+6y^{3}+x}$

Solutions

Logarithmic Differentiation

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

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

$y'=\sqrt[4]{1-x^{3}}-\frac{3x^{3}}{4(1-x^{3})^{3/4}}$

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

$y'=\frac{1}{2}\sqrt{\frac{x+1}{1-x}}\,(\frac{1}{x+1}+\frac{1}{1-x})$

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

$y'=(2x)^{2x}(2\ln(2x)+2)$

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

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

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

$y'=6x^{\cos(x)+1}(-\sin(x)\ln(x)+\frac{\cos(x)+1}{x})$

Solutions

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.

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

a) $x=0,-2$
b) $y=15x-22$

70. $f(x) = x^3 - 3x + 1, \;\;\; (1,-1)$

a) $x=\pm1$
b) $y=-1$

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

a) $x=2,-3$
b) $y=-12x+6$

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

a) $x=2^{-4/3}$
b) $y=\frac{3}{2}x+\frac{3}{2}$

73. $f(x) = (x^2+1)(2-x), \;\;\; (2,0)$

a) $x=1,\frac{1}{3}$
b) $y=-5x+10$

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

a) $x=-\frac{1}{2},-2$
/ b) $y=35x-\frac{115}{2}$

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

$y=\frac{2}{3}x-\frac{5}{3}$

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

$y=-2x+2$

Solutions

Higher Order Derivatives

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

$36x^2+6$

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

base case: Consider the zeroth-order polynomial, $c$. $\frac{dc}{dx}=0$
induction step: Suppose that the n-th derivative of a (n-1)th order polynomial is 0. Consider the n-th order polynomial, $f(x)$. We can write $f(x)=cx^n+P(x)$ where $P(x)$ is a (n-1)th polynomial.
$\frac{d^{n+1}}{dx^{n+1}}f(x)=\frac{d^{n+1}}{dx^{n+1}}(cx^n+P(x))=\frac{d^{n+1}}{dx^{n+1}}(cx^n)+\frac{d^{n+1}}{dx^{n+1}}P(x)=\frac{d^{n}}{dx^{n}}(cnx^{n-1})+\frac{d}{dx}\frac{d^{n}}{dx^{n}}P(x)=0+\frac{d}{dx}0=0$

Solutions