2.8 The Product Rule

Product Rule Formula

If h(x) = f(x)·g(x), then h'(x) = f'(x)g(x) + f(x)g'(x). You cannot just multiply the derivatives — the product rule accounts for both factors changing at once. Memorize it as: derivative of the first times the second, plus first times derivative of the second.

Why We Need It

When two functions are multiplied together, treating each factor separately captures how each one's rate of change contributes to the whole. This rule is essential for differentiating functions like x²·sin(x) or eˣ·ln(x), where the factors can't easily be simplified before differentiating.

Extending to Three Factors

For h(x) = f(x)g(x)k(x), apply the rule iteratively: h'(x) = f'gk + fg'k + fgk'. Each term differentiates one factor while leaving the others unchanged. This pattern generalizes the product rule to any number of factors multiplied together.

Key terms

Product Rule formula

h'(x) = f'(x)g(x) + f(x)g'(x), for h(x) = f(x)g(x).

Common mistake with products

You cannot simply multiply f'(x) and g'(x) — both terms of the product rule must be included.

Derivative of x²·sin(x)

2x·sin(x) + x²·cos(x)

Derivative of eˣ·ln(x)

eˣ·ln(x) + eˣ/x

Product Rule for three factors

h'(x) = f'gk + fg'k + fgk', applying the rule to each factor in turn.

Try an AP-style question

d/dx (x² sin x) = ?

1. 2x sin x + x² cos x
2. 2x sin x − x² cos x
3. 2x cos x
4. x² cos x