The Rules of Differentiation

In short: The rules of differentiation find derivatives without first principles. The power rule says $\frac{d}{d x} x^{n} = n x^{n - 1}$ , and the product, quotient and chain rules handle functions that are multiplied, divided or put inside each other.

Starting point: Years 9 and 10

The rules work on powers of x, so first rewrite everything as a power. This uses the index laws.

Written as	Rewrite as a power
$x$	$x^{\frac{1}{2}}$
$\frac{1}{x}$	$x^{- 1}$
$\frac{1}{x ^{3}}$	$x^{- 3}$
$\frac{5}{x}$	$5 x^{- \frac{1}{2}}$
$x x$	$x^{\frac{3}{2}}$
$\frac{x ^{2} + x}{x}$	$x + 1$

You also need two facts about straight lines:

The gradient of $y = 3 x + 2$ is 3, everywhere.
The gradient of $y = 7$ (a horizontal line) is 0.

These two facts are already derivatives: $\frac{d}{d x} (3 x + 2) = 3$ and $\frac{d}{d x} (7) = 0$ .

Differentiation rules

Power rule

Bring down, take one off
Constants vanish

Combining

Sum: term by term
Product: u'v + uv'
Quotient: (u'v - uv')/v²
Chain: outside times inside

Special functions

eˣ stays eˣ
ln x gives 1/x
sin gives cos
cos gives -sin
tan gives sec²

Year 12Derivatives of special functions

$f (x)$	$f^{'} (x)$
$e^{x}$	$e^{x}$
$e^{f (x)}$	$f^{'} (x) e^{f (x)}$
$ln x$	$\frac{1}{x}$
$ln f (x)$	$\frac{f ^{'} ( x )}{f ( x )}$
$sin x$	$cos x$
$cos x$	$- sin x$
$tan x$	$sec^{2} x$
$a^{x}$	$a^{x} ln a$

With the chain rule: $\frac{d}{d x} sin (3 x) = 3 cos (3 x)$ and $\frac{d}{d x} cos (x^{2}) = - 2 x sin (x^{2})$ .

Combining rules. $y = x^{2} e^{- x}$ needs the product rule:

$\frac{d y}{d x} = 2 x e^{- x} + x^{2} (- e^{- x}) = x e^{- x} (2 - x)$

Angles must be in radians for the trig derivatives.

Use log laws first. $y = ln \frac{x ^{3}}{x + 2} = 3 ln x - ln (x + 2)$ , so $\frac{d y}{d x} = \frac{3}{x} - \frac{1}{x + 2}$ . Much easier than the quotient rule inside a log.

Logs of any base. $lo g_{a} x = \frac{ln x}{ln a}$ , so $\frac{d}{d x} lo g_{a} x = \frac{1}{x ln a}$ . The tangent to $y = lo g_{2} x$ at $x = 2$ has gradient $\frac{1}{2 ln 2}$ .

The chain rule with function notation. If $y = f (3 x - 1)$ , then $\frac{d y}{d x} = 3 f^{'} (3 x - 1)$ . If $g (x) = [f (x)]^{2}$ , then $g^{'} (x) = 2 f (x) f^{'} (x)$ . So if $f (1) = 4$ and $f^{'} (1) = - 2$ , then $g^{'} (1) = 2 (4) (- 2) = - 16$ .

The second derivative. Differentiating twice gives $f^{''} (x)$ , also written $\frac{d ^{2} y}{d x ^{2}}$ . For $y = e^{2 x}$ , $y^{'} = 2 e^{2 x}$ and $y^{''} = 4 e^{2 x}$ . It measures how the gradient is changing, and is used in curve sketching.

A useful link. $\frac{d}{d x} tan x = sec^{2} x = 1 + tan^{2} x$ , so $y = tan x$ satisfies $\frac{d y}{d x} = 1 + y^{2}$ .

Quick summary

Rewrite roots and fractions as powers of $x$ before differentiating.
Power rule: $\frac{d}{d x} x^{n} = n x^{n - 1}$ . Constants on their own vanish.
Product: $u^{'} v + u v^{'}$ . Quotient: $\frac{u ^{'} v - u v ^{'}}{v ^{2}}$ . Chain: outside derivative times inside derivative.
$e^{x} \to e^{x}$ , $ln x \to \frac{1}{x}$ , $sin x \to cos x$ , $cos x \to - sin x$ , $tan x \to sec^{2} x$ .

Quiz

14 questions with instant marking. It is free, and your scores are saved to your account so you can carry on from any device.

Turn on JavaScript to sign in and take the quiz.

Try 3 questions now

Write $x$ as a power of $x$ .
1. A $x^{2}$
2. B $x^{\frac{1}{2}}$
3. C $x^{- \frac{1}{2}}$
4. D $x^{- 2}$
Show answer
B: $x^{\frac{1}{2}}$
A square root is the power one half.
Write $\frac{1}{x ^{4}}$ as a power of $x$ .
1. A $x^{4}$
2. B $x^{\frac{1}{4}}$
3. C $x^{- 4}$
4. D $- 4 x$
Show answer
C: $x^{- 4}$
A power on the bottom becomes negative.
Write $\frac{3}{x}$ as a power of $x$ .
1. A $3 x^{- \frac{1}{2}}$
2. B $3 x^{\frac{1}{2}}$
3. C $\frac{1}{3} x^{- \frac{1}{2}}$
4. D $3 x^{- 2}$
Show answer
A: $3 x^{- \frac{1}{2}}$
$x = x^{\frac{1}{2}}$ on the bottom becomes $x^{- \frac{1}{2}}$ .

Starting point: Years 9 and 10

Quick summary

Quiz

Feedback on this chapter

Mathematics Advanced chapters