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 Differentiation

Edexcel A level Maths: 9.3 The Chain 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 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 $\dfrac{dy}{dx} = \dfrac{dy}{du} \times \dfrac{du}{dx}$ . Differentiate the outer function, then multiply by the derivative of the inside.

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

Chain rule.

Example 1Differentiate

y = (2x + 1)^5

1Let

u = 2x+1

y = u^5

\tfrac{dy}{du} = 5u^4

\tfrac{du}{dx} = 2

\tfrac{dy}{dx} = 5(2x+1)^4 \cdot 2 = 10(2x+1)^4

Answer

10(2x+1)^4

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

A useful corollary

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

Formula recap

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

Chain rule.

\frac{dy}{dx} = \frac{1}{dx/dy}

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

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

The chain rule states dy/dx =
- $\frac{dy}{du}\cdot\frac{du}{dx}$
- $\frac{dy}{du}+\frac{du}{dx}$
- $\frac{du}{dy}\cdot\frac{dx}{du}$
- $\frac{dy}{dx}$
Show answer and explanation
Answer: $\frac{dy}{du}\cdot\frac{du}{dx}$
Product of derivatives.
$\frac{d}{dx}(2x+1)^5$ equals:
- $10(2x+1)^4$
- $5(2x+1)^4$
- $(2x+1)^4$
- $10(2x+1)^5$
Show answer and explanation
Answer: $10(2x+1)^4$
5u⁴·2.
$\frac{d}{dx}\,e^{x^2}$ equals:
- $2x e^{x^2}$
- $e^{x^2}$
- $x^2 e^{x^2}$
- $2x e^{2x}$
Show answer and explanation
Answer: $2x e^{x^2}$
e^u·u′ with u=x².
The reciprocal form is dy/dx =
- $\frac{1}{dx/dy}$
- $dx/dy$
- $\frac{dy}{dx}$
- $1$
Show answer and explanation
Answer: $\frac{1}{dx/dy}$
Useful when x(y) is simpler.
$\frac{d}{dx}\sin(3x)$ equals:
- $3\cos(3x)$
- $\cos(3x)$
- $-3\cos(3x)$
- $3\sin(3x)$
Show answer and explanation
Answer: $3\cos(3x)$
cos(3x)·3.

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