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:

\[ \frac {d}{dx} \left (f(x)\cdot g(x)\right ) = f'(x) \cdot g'(x)? \]

Devyn

Hmmm, let’s give this theory an acid test. Let’s try

\[ f(x) = x^2+1\qquad \text {and}\qquad g(x) = x^3-3x \]

Now \begin{align*} f'(x)g'(x) &= (2x)(3x^2-3)\\ &= 6x^3-6x. \end{align*}

Riley

On the other hand, \begin{align*} f(x)g(x) &= (x^2+1)(x^3-3x)\\ &=x^5-3x^3+x^3-3x\\ &=x^5-2x^3-3x. \end{align*}

Devyn

And so,

\[ \frac {d}{dx} \left (f(x) \cdot g(x)\right ) = 5x^4-6x^2-3. \]

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.