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MathematicsA Level & IB

What is chain rule?

The chain rule differentiates a composite function — a function inside another function — by multiplying the derivatives of the outer and inner parts.

Chain rule explained

Write the inside as uu, differentiate the outer function with respect to uu, then multiply by dudx\frac{du}{dx}. It is the single most used rule in A Level differentiation.
It also drives connected rates of change, where you chain derivatives together to link quantities such as radius, area and time.

Key formula

dydx=dydu×dudx\dfrac{dy}{dx} = \dfrac{dy}{du} \times \dfrac{du}{dx}

Worked example

For y=(3x+1)5y = (3x + 1)^{5}, let u=3x+1u = 3x + 1: dydu=5u4\frac{dy}{du} = 5u^{4} and dudx=3\frac{du}{dx} = 3, so dydx=15(3x+1)4\frac{dy}{dx} = 15(3x + 1)^{4}.

Examiner tip

Show the substitution line. Writing down uu and both derivatives protects the method mark even if you mis-simplify.

Chain rule: common questions

When do I use the chain rule instead of the product rule?
Use the chain rule for a function inside a function, such as $\sin(2x)$, and the product rule for two functions multiplied together, such as $x\sin x$.
Can I use the chain rule with trigonometric functions?
Yes — $\frac{d}{dx}\sin(kx) = k\cos(kx)$ follows directly from it, and the same pattern works for exponentials and logarithms.

Related terms

Differentiation

Differentiation is the process of finding the rate at which one quantity changes with respect to another — the gradient of a curve at a point.

Product rule

The product rule differentiates the product of two functions: differentiate the first times the second, plus the first times the derivative of the second.

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