9.3PureStretch

The Chain Rule

The chain rule differentiates a composite function — a function of a function. It is the most-used differentiation rule in Year 2, underpinning the product, quotient, parametric and implicit methods.

26 min Video by Zeeshan Zamurred Differentiation

Edexcel A level Maths: 9.3 The Chain RuleWatch the full walkthrough before the notes below.

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What you'll be able to do

State and apply the chain rule
Differentiate composite functions
Use the dy/dx = dy/du · du/dx form
Differentiate powers of functions quickly

The rule

If $y$ is a function of $u$ and $u$ is a function of $x$ , then $\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}$ . Differentiate the outer function, then multiply by the derivative of the inside.

\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}

Chain rule.

1Let

u = 2 x + 1

y = u^{5}

\frac{d y}{d u} = 5 u^{4}

\frac{d u}{d x} = 2

\frac{d y}{d x} = 5 (2 x + 1)^{4} \cdot 2 = 10 (2 x + 1)^{4}

Answer

10 (2 x + 1)^{4}

Tip — “Differentiate the outside, keep the inside, times the derivative of the inside.”

A useful corollary

A related result: $\frac{d y}{d x} = \frac{1}{d x / d y}$ , handy when $x$ is easier to express in terms of $y$ .

Formula recap

\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}

Chain rule.

\frac{d y}{d x} = \frac{1}{d x / d y}

Reciprocal form.

Common mistakes to avoid

Forgetting to multiply by the derivative of the inside.

Always times du/dx.

Differentiating the inside function as well as the outside in one step incorrectly.

Outer derivative × inner derivative, kept separate.

Key takeaways

dy/dx = dy/du · du/dx.
Differentiate outside, keep inside, × derivative of inside.
dy/dx = 1/(dx/dy) when convenient.

Test yourself

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