Calculus/Differentiation/Basics of Differentiation/Exercises

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Differentiation/Basics of Differentiation/Exercises

Find the Derivative by Definition[edit]

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[edit]

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[edit]

Find the derivative of the following functions:

Power Rule[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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

External Links[edit]


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Differentiation/Basics of Differentiation/Exercises