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.