Product and Quotient Rules

The product rule is simply the recipe for differentiating two functions multiplied together, and the quotient rule does the same for one divided by another. You need them because, as we saw last lesson, the derivative of a product is not the product of the derivatives. The product rule $(fg)' = f'g + fg'$ is just two pieces added up: $f'g$ (the change you get from $f$ moving while $g$ holds still) and $fg'$ (the change from $g$ moving while $f$ holds still). The quotient rule looks worse, but it's the same two pieces with a minus between them, all over $g^2$ (the bottom, squared). Bear with it; there's a picture that makes the product rule obvious.

Why the product rule looks like that. Picture a rectangle with width $f$ and height $g$, so its area is $fg$. Now let both sides grow a little. The area picks up three new pieces: a strip along one side (the width grew by a bit while the height stayed $g$, that's $f'g$), a strip along the other (the height grew while the width stayed $f$, that's $fg'$), and a tiny corner where both grew at once. That corner is a small change times a small change, so it's far smaller than the strips and vanishes in the limit. What's left is $f'g + fg'$. There's an animation further down that shows the strips growing.

The quotient rule. People remember it as "low d-high minus high d-low, over the square of what's below." Unlike the product rule, the order matters here because of the minus sign: the derivative of the top comes first.

Simplify first when you can. $\dfrac{x^3 + 2x}{x}$ is just $x^2 + 2$, so there's no need for the quotient rule at all.

The same thinking extends to more factors. With three, each factor takes one turn being differentiated while the others hold still: $(fgh)' = f'gh + fg'h + fgh'$.

Revenue is price × quantity, and both change with time, so the rate of change of revenue is a product-rule calculation. Concentration is amount ÷ volume, a quotient-rule one.