Calculus/Differentiation/Basics of Differentiation/Solutions

Find the Derivative by DefinitionEdit

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Prove the Constant RuleEdit

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

Find the Derivative by RulesEdit

Power RuleEdit

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Product RuleEdit

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Quotient RuleEdit

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Chain RuleEdit

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Let  . Then
 
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Let  . Then

 
 
 

 
Let  . Then

 
 
 

 
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Let  . Then

 
 
 

 
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Let  . Then

 
 
 

 
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Let  . Then

 
 
 

 
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Let  . Then
 
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ExponentialsEdit

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Let  . Then
 
Let  . Then
 
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Let
 
 
 

Then

 

Using the chain rule, we have

 

The individual factor are

 
 
 

So

 
Let
 
 
 

Then

 

Using the chain rule, we have

 

The individual factor are

 
 
 

So

 
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LogarithmsEdit

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Let  . Then
 
Let  . Then
 
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Trigonometric functionsEdit

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More DifferentiationEdit

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Let  . Then

 
 

 
Let  . Then

 
 

 
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Implicit DifferentiationEdit

Use implicit differentiation to find y'

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Logarithmic DifferentiationEdit

Use logarithmic differentiation to find  :

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