mathsGrade 11-122 min read

Differentiation: the five rules you actually need

Power, sum, product, quotient, chain — the whole toolkit on one page, with the one question to ask before you pick a rule.

Differentiation feels like a long list of rules. It isn't. It's five, and most exam questions need only the first two. The skill isn't memorising more — it's reading the function and asking one question: what is this, structurally?

The one question

Before you differentiate anything, name its shape:

A single term raised to a power? → power rule.
Several terms added together? → sum rule, then power rule on each.
Two functions multiplied? → product rule.
One function divided by another? → quotient rule.
A function inside another function? → chain rule.

Get the shape right and the rule chooses itself.

1. Power rule

\frac{d}{dx}\left(x^n\right) = n\,x^{\,n-1}

Bring the power down, drop it by one. Works for negative and fractional powers too: $\frac{d}{dx}\!\left(\sqrt{x}\right) = \frac{d}{dx}\!\left(x^{1/2}\right) = \tfrac{1}{2}x^{-1/2}$ .

2. Sum rule

\frac{d}{dx}\big(f \pm g\big) = f' \pm g'

Differentiate term by term. Constants vanish; a constant times a function keeps the constant: $\frac{d}{dx}(5x^3) = 15x^2$ .

3. Product rule

\frac{d}{dx}\big(u\,v\big) = u'v + uv'

Say it out loud: "first times derivative of the second, plus second times derivative of the first." The rhythm stops you dropping a term.

4. Quotient rule

\frac{d}{dx}\!\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}

The order matters here — it's $u'v$ minus $uv'$ , not the other way round. Swap them and the sign is wrong.

5. Chain rule

\frac{d}{dx}\,f\big(g(x)\big) = f'\big(g(x)\big)\cdot g'(x)

Differentiate the outside, leave the inside alone, then multiply by the derivative of the inside. "Outside, then in."

Worked example

Differentiate $y = (3x^2 + 1)^4$ .

This is a function inside a function → chain rule. Outside is $(\,\cdot\,)^4$ , inside is $3x^2 + 1$ .

\frac{dy}{dx} = 4(3x^2 + 1)^3 \cdot 6x = 24x\,(3x^2 + 1)^3

Exam tip: before you touch a pencil, write the rule's name in the margin. Naming the shape first turns "I don't know where to start" into a one-line decision — every time.

Last revised 10 May 2026.