Product Rule

 

Product Rule

     If f and g are two differentiable functions, then

 

      

 

Let us begin with the definition of a derivative.

 

 

 

The key is to subtract and add a term: . You need to know to do this to make any progress. Doing this, we get the following:

 

.

 

From the property of limits, we can break the limit into two pieces because the limit of a sum is the sum of the limits. And so, we have:

 

 

 

Factoring a  on the first limit and  from the second limit we get:

 

 

 

Another property of limits says that the limit of a product is a product of the limits. Using this fact, we can rewrite the limit as:

 

 

 

By definition, though,  and .

 

Also,  and , since they do not depend on h.

 

So, we have , which is what we wanted to show.