Cute Proofs: The Product Rule for Derivatives.

There’s always a few calculus students who make the error of trying to take the derivative of and get .  Of course, we know that this is not true in general, and the product rule for derivatives is as follows:

 

Theorem (Product Rule).  If are differentiable, then . 

 

Given this formula, it’s a nice exercise for students to find out for which functions it is true that .