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.  
 
 
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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

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Solutions

Product RuleEdit

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Solutions

Quotient RuleEdit

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Solutions

Chain RuleEdit

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Solutions

ExponentialsEdit

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Solutions

LogarithmsEdit

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Solutions

Trigonometric functionsEdit

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Solutions

More DifferentiationEdit

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Solutions

Implicit DifferentiationEdit

Use implicit differentiation to find y'

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Solutions

Logarithmic DifferentiationEdit

Use logarithmic differentiation to find  :

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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.

81.  
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b)  
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82.  
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84.  
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86.  
a)  
/ b)  
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/ b)  
87. Find an equation of the tangent line to the graph defined by   at the point (1,-1).
 
 
88. Find an equation of the tangent line to the graph defined by   at the point (1,0).
 
 

Solutions

Higher Order DerivativesEdit

89. What is the second derivative of  ?
 
 
90. 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.

 

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

Advanced Understanding of DerivativesEdit

91. Let   be the derivative of  . Prove the derivative of   is  .

Suppose  . Let  .
 

Therefore, if   is the derivative of  , then   is the derivative of  .  

Suppose  . Let  .
 

Therefore, if   is the derivative of  , then   is the derivative of  .  
92. Suppose a continuous function   has three roots on the interval of  . If  , then what is ONE true guarantee of   using
(a) the Intermediate Value Theorem;
(b) Rolle's Theorem;
(c) the Extreme Value Theorem.
These are examples only. More valid solutions may exist.
(a)   is continuous. Ergo, the intermediate value theorem applies. There exists some   such that  , where  .
(b) Rolle's Theorem does not apply for a non-differentiable function.
(c)   is continuous. Ergo, the extreme value theorem applies. There exists a   so that   for all  .
These are examples only. More valid solutions may exist.
(a)   is continuous. Ergo, the intermediate value theorem applies. There exists some   such that  , where  .
(b) Rolle's Theorem does not apply for a non-differentiable function.
(c)   is continuous. Ergo, the extreme value theorem applies. There exists a   so that   for all  .
93. Let  , where   is the inverse of  . Let   be differentiable. What is  ? Else, why can   not be determined?
 .
 .
94. Let   where   is a constant.

Find a value, if possible, for   that allows each of the following to be true. If not possible, prove that it cannot be done.

(a) The function   is continuous but non-differentiable.
(b) The function   is both continuous and differentiable.
(a)  .
(b) There is no   that allows the following to be true. Proof in solutions.
(a)  .
(b) There is no   that allows the following to be true. Proof in solutions.

Solutions

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Differentiation/Basics of Differentiation/Exercises