\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'

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