The chain rule differentiates composite functions by multiplying the derivative of the outer function by the derivative of the inner function.

What this dot point is asking

A composite function is a "function of a function" - one expression nested inside another, like $(3x+1)^5$, $e^{x^2}$ or $\ln(\cos x)$. The chain rule tells you how to differentiate these.

The rule

If $y=f(u)$ and $u=g(x)$, then

Equivalently, for $y=f(g(x))$,

The intuition: differentiate the outside function treating the inside as a single block, then multiply by the derivative of the inside.

Standard chain-rule results

These come up constantly and are worth knowing by sight:

• $\dfrac{d}{dx}\big[g(x)\big]^n=n\big[g(x)\big]^{n-1}\,g'(x)$
• $\dfrac{d}{dx}e^{g(x)}=g'(x)\,e^{g(x)}$
• $\dfrac{d}{dx}\ln\big(g(x)\big)=\dfrac{g'(x)}{g(x)}$
• $\dfrac{d}{dx}\sin\big(g(x)\big)=g'(x)\cos\big(g(x)\big)$

Combining with other rules

In practice the chain rule rarely appears alone. Watch for it inside products and quotients.

For $y=x\,e^{3x}$, use the product rule with $u=x$ and $v=e^{3x}$, where $v'=3e^{3x}$ comes from the chain rule:

Common errors

Why it matters

The chain rule unlocks differentiation of exponential and logarithmic models, which dominate Topic 4, and it is essential for optimisation problems where a quantity is expressed through an intermediate variable. Combined with the product and quotient rules, it lets you differentiate essentially any function you will meet in Stage 2.