9.4PureStretch

The Product Rule

The product rule differentiates a product of two functions. You cannot just multiply the separate derivatives — each term keeps one factor and differentiates the other.

24 min Differentiation

Edexcel A level Maths: 9.4 and 9.5 The Product and Quotient RuleCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

State and apply the product rule
Differentiate products like x²eˣ
Combine with the chain rule
Simplify and factorise the result

The rule

If $y = uv$ where $u$ and $v$ are functions of $x$ , then $\frac{d y}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x}$ , often written $u^{'} v + u v^{'}$ .

\frac{d}{d x} (uv) = u \frac{d v}{d x} + v \frac{d u}{d x}

Product rule.

u = x^{2}

v = e^{x}

u^{'} = 2 x

v^{'} = e^{x}

\frac{d y}{d x} = x^{2} e^{x} + e^{x} (2 x) = e^{x} (x^{2} + 2 x)

Answer

e^{x} (x^{2} + 2 x)

Tip — Differentiate one, keep the other; add the two terms. Factor the answer where possible.

Formula recap

\frac{d}{d x} (uv) = u^{'} v + u v^{'}

Product rule.

Common mistakes to avoid

Writing (uv)′ = u′v′.

It is u′v + uv′.

Forgetting the chain rule on a composite factor.

Differentiate each factor correctly, using the chain rule if needed.

Key takeaways

(uv)′ = u′v + uv′.
Differentiate one factor, keep the other, then add.
Factor the result to simplify.

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