Product Rule
Product RuleIf 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.