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Products and Quotients

The Quotient Rule

Let us first consider the derivative of the following quotient:

          

Now consider the quotient of the following derivatives:

          

From the above we can see the following important point:

          

 

The derivative of a quotient is NOT the quotient of the derivatives!

 

 

To find the derivative of a quotient we use the following quotient rule:

Quotient Rule:

 

 

In words we say: The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all divided by the denominator squared.

Click here to see a proof of the quotient rule.

 

Example

Click on the question marks to see the following example done step-by-step:









Note that the numerator in the quotient rule looks very similar to the product rule. The only difference is that we subtract f(x)g'(x) from g(x)f'(x) rather than add it. Keeping this in mind, there is a visual method for using the quotient rule that is very similar to that for the product rule and can be used as an alternative to memorizing the formula for the quotient rule.

 

to see the visual method. Before clicking, make sure all of the white space below is in your viewing window (scroll down if necessary).

  

 

 

 

Example

Find the derivative of

to see the visual method. Before clicking, make sure all of the white space below is in your viewing window (scroll down if necessary).

  

 

 

Exercise

Try solving these exercises using the quotient rule.

Exercise

Find the derivative of

f(x) =



Working:

f '(x) =



<< The Product Rule  |  Differentiation Index |  Chain Rule >>

 

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