AP Calculus AB and BC

Product Rule vs Chain Rule

Use the product rule when two functions are multiplied side by side, and the chain rule when one function sits inside another. The test is to ask what you would compute last if you evaluated by hand: a multiplication means product rule, an outer function means chain rule.

Product rule

Use when: Two factors are multiplied and both contain the variable, as in $x^2 \sin x$ .

Chain rule

Use when: One function is evaluated inside another, as in $\sin(x^2)$ .

Side by side

	Product rule	Chain rule
Trigger	$f(x) \cdot g(x)$	$f(g(x))$
Formula	$f g' + g f'$	$f'(g(x)) \cdot g'(x)$
Example	$x^2 \sin x$	$\sin(x^2)$
Result	$x^2\cos x + 2x\sin x$	$2x\cos(x^2)$

The two examples in the table differ only by where the parentheses sit, and that changes the answer completely. In $x^2 \sin x$ nothing is fed into anything, so the operations are independent. In $\sin(x^2)$ the squaring happens first and its output is handed to the sine.

Real problems often need both. Differentiating $x^2 \sin(3x)$ starts with the product rule, and the chain rule then handles $\sin(3x)$ inside it. Deciding the outermost structure first keeps the work organized.

The mistake

Applying the chain rule to a product by differentiating the outer factor and multiplying by the derivative of the inner one. That is not a rule, and it gives the wrong answer for $x^2 \sin x$ .

Frequently asked questions

How do I know which rule is on the outside?

Imagine plugging in a number and evaluating by hand. Whatever operation you would perform last is the outermost structure, and that decides the first rule you apply.

In the CED: Unit 2: Defining the Derivative, Unit 3: Chain Rule, Implicit, and Inverses