Product Rule vs. Chain Rule: How to Tell Which One You Need

\frac{d}{d x} [f (x) g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x)

\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) g^{'} (x)

How to tell them apart

Imagine evaluating the function. If the last thing you do is multiply two pieces that contain $x$ , start with the product rule. If the last thing you do is apply an outer function to an inner expression, start with the chain rule.

Product: $x^{2} sin x$

$f = x^{2}$ and $g = sin x$

$\frac{d}{d x} = 2 x sin x + x^{2} cos x$

Chain: $sin (x^{2})$

Outer function $sin$ , inner function $x^{2}$

$\frac{d}{d x} = 2 x cos (x^{2})$

Both: $x^{2} sin (3 x)$

Use the product rule, and the chain rule on $sin (3 x)$ .

$\frac{d}{d x} = 2 x sin (3 x) + 3 x^{2} cos (3 x)$

Common mistake

$\frac{d}{d x} (x^{2} + 1)^{5} = 5 (x^{2} + 1)^{4}$

This forgets the derivative of the inside. The correct answer is $10 x (x^{2} + 1)^{4}$ .

Try it yourself

$x e^{2 x}$
Show answer
$e^{2 x} + 2 x e^{2 x}$
$cos (5 x)$
Show answer
$- 5 sin (5 x)$
$x^{3} ln x$
Show answer
$3 x^{2} ln x + x^{2}$