Calculus/Differentiation/Basics of Differentiation/Exercises

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Differentiation/Basics of Differentiation/Exercises

Find the Derivative by DefinitionEdit

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

1.  

 

2.  

 

3.  

 

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9.  

 

Solutions

Prove the Constant RuleEdit

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

 

Solutions

Find the Derivative by RulesEdit

Find the derivative of the following functions:

Power RuleEdit

11.  

 

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Solutions

Product RuleEdit

20.  

 

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25.  

 

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28.  

 

Solutions

Quotient RuleEdit

24.  

 

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Solutions

Chain RuleEdit

31.  

 

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Solutions

ExponentialsEdit

42.  

 

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Solutions

LogarithmsEdit

46.  

 

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Solutions

Trigonometric functionsEdit

51.  

 

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Solutions

More DifferentiationEdit

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Solutions

Implicit DifferentiationEdit

Use implicit differentiation to find y'

62.  

 

63.  

 

Solutions

Logarithmic DifferentiationEdit

Use logarithmic differentiation to find  :

64.  

 

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66.  

 

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68.  

 

Solutions

Equation of Tangent LineEdit

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 DerivativesEdit

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

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Differentiation/Basics of Differentiation/Exercises