The Product and Quotient Rules — CAIE A-Level Mathematics

Introduction

Many functions encountered in A-Level Mathematics cannot be differentiated by simple rules applied to a single function — they are built by multiplying or dividing two distinct functions together. Examples such as $x^2 \ln x$ , $x e^{1-x^2}$ , and $\dfrac{2x-4}{3x^2+2}$ appear regularly in 9709 Paper 3 examinations. The Product Rule and Quotient Rule are the two essential techniques that handle these cases systematically. Mastery of both rules — including knowing which to apply and how to simplify the result — is a core differentiable skill for Paper 3.

Core Concept

The Product Rule

When a function can be written as $y = u(x)\,v(x)$ — a product of two differentiable functions — its derivative is not simply $u'v'$ . Instead:

\frac{\mathrm{d}y}{\mathrm{d}x} = u\frac{\mathrm{d}v}{\mathrm{d}x} + v\frac{\mathrm{d}u}{\mathrm{d}x}

In words: (first × derivative of second) + (second × derivative of first).

Identification: Look for two distinct function types multiplied together — e.g. a power of $x$ multiplied by $e^x$ , $\ln x$ , $\sin x$ , etc.

The Quotient Rule

When $y = \dfrac{u(x)}{v(x)}$ — a quotient of two differentiable functions — the derivative is:

\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{v\dfrac{\mathrm{d}u}{\mathrm{d}x} - u\dfrac{\mathrm{d}v}{\mathrm{d}x}}{v^2}

In words: (bottom × derivative of top) − (top × derivative of bottom), all over (bottom)².

Identification: A fraction in which both numerator and denominator contain $x$ (and cannot be trivially simplified to a single standard function).

Note: An alternative to the Quotient Rule is to rewrite $\dfrac{u}{v}$ as $u \cdot v^{-1}$ and apply the Product Rule combined with the Chain Rule. Both methods are equally valid in examinations.

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Introduction

Core Concept

The Product Rule

The Quotient Rule

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