Chain rule: find the inside function first

When a chain rule question goes wrong, it usually goes wrong before any differentiating starts. You can't apply the rule until you know what the inside function is.

Name the inside first

Take \(y = (x^2 - 4)^3\). Before you write anything else, write the inside:

\[ u = x^2 - 4, \qquad \frac{du}{dx} = 2x. \]

Now the outside is just \(u^3\), which you already know how to differentiate. Multiply the two pieces:

\[ \frac{dy}{dx} = 3u^2 \cdot 2x = 6x(x^2 - 4)^2. \]

The three slips that cost marks

1. Forgetting the inside derivative. \(\frac{d}{dx}\sin(4x)\) is \(4\cos(4x)\), not \(\cos(4x)\).
2. Changing the inside. The outside derivative keeps the original inside: \(\frac{d}{dx}e^{3x^2} = 6x\,e^{3x^2}\), not \(6x\,e^{6x}\).
3. Missing a hidden chain. \(\sin^2 x\) is \((\sin x)^2\), so it needs the chain rule too.

Practise it

The free practice set below has twelve calculator-free questions with full worked solutions. Do the first six without looking, then check every line of your working, not just the answers.