Choose the rule before calculating.

The chain rule differentiates a composition: take the derivative of the outside, keeping the inside, then multiply by the derivative of the inside. For example, ( $2x$ + 1)³ differentiates to 6( $2x$ + 1)².

Products and quotients require their own rules. Sometimes expanding or simplifying first is easier, but preserve domain restrictions. Several algebraic forms of a derivative can be equivalent; compare their meaning rather than their appearance.

INTERACTIVE MODEL

Try it. Watch it change.

f(x)

x^{3}

− 3xtangent

f'(x)

FunctionTangent point x1

f′(1) = 0 · f″(1) = 6

The graph is calculated from the displayed rule. Drag the highlighted point or use the sliders.

Explore: Switch function families and move the tangent point. Compare the stated derivative with its numerical gradient.

Product rule

(uv)'=u'v+uv'

Quotient rule

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

WORKED EXAMPLE

Differentiate y = $x^{2}$ (x + 1)³.

Use the product rule: y′ = $2x$ (x + 1)³ + $x^{2}$ ·3(x + 1)².
Factor common terms: y′ = x(x + 1)²(5x + 2).

Assumed knowledge

Power rule and factorisation.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Apply chain, product and quotient rules to algebraic functions.
Combine rules and check equivalent derivative forms.

QCAA Mathematical Methods 2025 v1.3 · p. 22 ↗

READY TO TRY IT?

Choose the rule before calculating.

Differentiate y = x2x^{2}x2(x + 1)³.

Put the idea to work.

Differentiate y = $x^{2}$ (x + 1)³.