
Two young mathematicians discuss derivatives of products and products of derivatives.

Check out this dialogue between two calculus students (based on a true story):
Devyn
Hey Riley, remember the sum rule for derivatives?
Riley
You know I do.
Devyn
What do you think that the “product rule” will be?
Riley
Let’s give this a spin:
Devyn
Hmmm, let’s give this theory an acid test. Let’s try Now
Riley
On the other hand,
Devyn
And so,
Riley
Wow. Hmmm. It looks like our guess was incorrect.
Devyn
I’ve got a feeling that the so-called “product rule” might be a bit tricky.
Above, our intrepid young mathematicians guess that the “product rule” might be: Does this ever hold true?
Answers will vary. A partial answer is that this will hold when either $f(x)$ or $g(x)$ are zero, or when both are constants.