Product Rule

(fg)' = f'g + fg' — differentiate each factor, swap, add.

The idea

Theorem (Product rule).

If $f$ and $g$ are differentiable, then $(fg)' = f'g + fg'.$

The rule has two terms because a product changes for two reasons: $f$ can change while $g$ holds still, and $g$ can change while $f$ holds still. Picture $fg$ as the area of a rectangle with sides $f$ and $g$, as in the figure. Widening the rectangle a little, by $\Delta f$, adds a thin strip of height $g$, which arrives at rate $f'g$; making it taller by $\Delta g$ adds a thin strip of width $f$, at rate $fg'$. The corner where both sides grow at once is the product $\Delta f \cdot \Delta g$ of two small changes, and it vanishes in the limit.

The derivative of a product is not the product of the derivatives. Take $f(x) = x$ and $g(x) = x$: the product is $x^{2}$, whose derivative is $2x$, while the product of the derivatives is $1 \cdot 1 = 1$. Differentiation passes through sums, but not through products.

To apply the rule, differentiate each factor in turn, multiply by the other factor left alone, and add the two results.

Ways to work on it

• Walkthrough. Apply the product rule to a simple product and verify it by expanding first.
• Proof. Derive the product rule from the limit definition with the add-and-subtract trick.
• Practice. Differentiate x^a(x + b) for random (a, b).
• Hardest. Differentiate x^n times a non-polynomial ( , , e^x) — the product rule is unavoidable.

Not sure where to start? Take the ten-question placement test.