Calculus/Differentiation/Basics of Differentiation/Exercises

From Wikibooks, open books for an open world
Jump to: navigation, search
← Some Important Theorems Calculus L'Hôpital's rule →
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.

2.

3.

4.

5.

6.

7.

8.

9.

Solutions

Prove the Constant Rule[edit]

10. Use the definition of the derivative to prove that for any fixed real number ,

Solutions

Find the Derivative by Rules[edit]

Find the derivative of the following functions:

Power Rule[edit]

11.

12.

13.

14.

15.

16.

17.

18.

19.

Solutions

Product Rule[edit]

20.

21.

22.

23.

Solutions

Quotient Rule[edit]

24.

25.

26.

27.

28.

29.

30.

Solutions

Chain Rule[edit]

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

Solutions

Exponentials[edit]

42.

43.

44.

45.

Solutions

Logarithms[edit]

46.

47.

48.

49.

50.

Solutions

Trigonometric functions[edit]

51.

52.

Solutions

More Differentiation[edit]

53.

54.

55.

56.

57.

58.

59.

60.

61.

Solutions

Implicit Differentiation[edit]

Use implicit differentiation to find y'

62.

63.

Solutions

Logarithmic Differentiation[edit]

Use logarithmic differentiation to find :

64.

65.

66.

67.

68.

Solutions

Equation of Tangent Line[edit]

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

69.

a)
b)

70.

a)
b)

71.

a)
b)

72.

a)
b)

73.

a)
b)

74.

a)
/ b)

75. Find an equation of the tangent line to the graph defined by at the point (1,-1).

76. Find an equation of the tangent line to the graph defined by at the point (1,0).

Solutions

Higher Order Derivatives[edit]

77. What is the second derivative of ?

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, .
induction step: Suppose that the n-th derivative of a (n-1)th order polynomial is 0. Consider the n-th order polynomial, . We can write where is a (n-1)th polynomial.

Solutions

External Links[edit]


← Some Important Theorems Calculus L'Hôpital's rule →
Differentiation/Basics of Differentiation/Exercises