## Cute Proofs: The Product Rule for Derivatives.

### November 3, 2010

$\mbox{\tiny Two functions (solid blue and red) make the dotted purple function when multiplied!}$

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

Theorem (Product Rule).  If $f, g$ are differentiable, then $(fg)' = f'g + g'f$

Given this formula, it’s a nice exercise for students to find out for which functions it is true that $(fg)' = f'g'$