The chain rule

Covers AS91578

What the chain rule is for

A composite function is a "function inside a function", like $(3x^2 + 1)^5$ — the inside is $3x^2 + 1$ , and the outside is "raise to the power 5".
The chain rule differentiates a composite function:

$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$

In words: differentiate the outside (leaving the inside alone), then multiply by the derivative of the inside.

Using the chain rule quickly

For $y = \big(f(x)\big)^n$ , the rule gives:

$\frac{dy}{dx} = n\big(f(x)\big)^{n-1} \times f'(x)$

Bring the power down, reduce the power by one, then multiply by the derivative of the bracket.

Worked ExampleDifferentiating a bracket to a power

Differentiate $y = (3x^2 + 1)^5$ .

Step 1 — Differentiate the outside, keeping the inside

Bring the 5 down and reduce the power to 4:

$5(3x^2 + 1)^4$

Step 2 — Multiply by the derivative of the inside

The inside is $3x^2 + 1$ , whose derivative is $6x$ :

$\frac{dy}{dx} = 5(3x^2 + 1)^4 \times 6x = 30x(3x^2 + 1)^4$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Differentiate $y = (2x + 1)^4$ .

Merit

Differentiate $y = \sqrt{x^2 + 5}$ .

Excellence

Find the gradient of $y = (x^2 - 3x)^4$ at the point where $x = 1$ .