Derivatives of Standard Functions — Pure Mathematics 3 (9709) — Mathematics

Introduction

Differentiation of standard functions is a core skill tested throughout Paper 3. The six standard derivatives — $e^x$ , $\ln x$ , $\sin x$ , $\cos x$ , $\tan x$ , and $\tan^{-1} x$ — form the building blocks for virtually every differentiation question at A-Level. You will need to combine these with constant multiples, sums and differences, and especially composite functions (the chain rule). Mastery here unlocks implicit differentiation, parametric differentiation, integration by recognition, and much more.

Core Concept

The Six Standard Derivatives

Every derivative below must be known without derivation in the exam.

Function $f(x)$	Derivative $f'(x)$
$e^x$	$e^x$
$\ln x$	$\dfrac{1}{x}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\tan^{-1} x$	$\dfrac{1}{1+x^2}$

Note: All trigonometric derivatives assume $x$ is measured in radians — this is always assumed in 9709.

Constant Multiples and Sums/Differences

These follow directly from the linearity of differentiation:

\frac{d}{dx}[af(x) + bg(x)] = af'(x) + bg'(x)

for constants $a, b$ .

Composite Functions — The Chain Rule

When a standard function is composed with another function $u(x)$ , apply the chain rule:

\frac{d}{dx}[f(u)] = f'(u)\cdot\frac{du}{dx}

This is the single most important technique for this objective. For example, $\sin(3x^2 + 1)$ is a composite: outer function $\sin(\cdot)$ , inner function $u = 3x^2 + 1$ .

Extended Standard Forms

By applying the chain rule to each standard function with a general inner function $u = g(x)$ :

Function	Derivative
$e^{g(x)}$	$g'(x)\,e^{g(x)}$
$\ln(g(x))$	$\dfrac{g'(x)}{g(x)}$
$\sin(g(x))$	$g'(x)\cos(g(x))$
$\cos(g(x))$	$-g'(x)\sin(g(x))$
$\tan(g(x))$	$g'(x)\sec^2(g(x))$
$\tan^{-1}(g(x))$	$\dfrac{g'(x)}{1+[g(x)]^2}$

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