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Products and Quotients

Background

The product rule provides us with a method for finding the derivative of a product of two functions. The quotient rule provides us with a method for finding the derivative of a quotient of two functions. Before learning either of these two rules we must first have a good understanding of how to find the derivative of the basic functions (f(x) = c, where c is a constant, xn, ln x, ex, sin x and cos x), how to use the multiple rule to find the derivatives of a constant times any of these basic functions, and how to use the sum and difference rules to find the derivatives of the sums or differences of any of these basic functions and their constant multiples.



The Product Rule

Let us first consider the derivative of the following product:

          

Now consider the product of the following derivatives:

          

From the above we can see the following important point:

          

 

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

 

 

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

Product Rule:

 

 

In words we say: The derivative of a product is the second times the derivative of the first plus the first times the derivative of the second.

Click here to see a proof of the product rule.

Example 1A:

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








There is an easier way to solve the above problem. If we multiply the functions first and then find the derivative we obtain:

                    

You can see that by either method we get the same answer. In many cases, however, the product rule is the only option. For example, the following can only be found using the product rule:

                    

The following visual method can be used as an alternative to memorizing the formula for the product 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 1B:

Find the derivative of

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

  

 

 

 

 

Examples

Try solving these examples using the product rule.

Some people find the Product Rule easier to remember if it is written with the first multiplication in the reverse order to that above.

(fg)' = f'g+fg'

The two ways are totally equivalent. Choose one way of writing it down and stick to that way.

Exercise

Find the derivative of

f(x) =

Working:

f '(x) =

<< The Sum and Difference Rules  |  Differentiation Index |  The Quotient Rule>>

 

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