The Chain Rule

Differentiate a function inside a function: powers of brackets, roots, e, sin and cos of an expression, and (0606/4037) ln and tan of an expression.

In the exam for IGCSE Additional Maths 0606, Edexcel Further Pure Maths 4PM1 · 4 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

A function inside a function

Look at $y = (3 x + 1)^{5}$ . You could expand the bracket, but that is five brackets multiplied out. There is a faster way.

This curve is built in two stages. First x goes into the bracket: $u = 3 x + 1$ . Then the bracket is raised to the power 5: $y = u^{5}$ . One function sits inside another.

Example

Think of gears. If gear A turns 5 times for each turn of gear B, and gear B turns 3 times for each turn of gear C, then A turns 5 × 3 = 15 times for each turn of C. The rates multiply.

Formula

\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}

$u$ — the inside part, for example 3x + 1.
$\frac{d y}{d u}$ — the gradient of the outside, treating u as the letter.
$\frac{d u}{d x}$ — the gradient of the inside.

Rule

Learn it. 0606 and 4037 give no calculus formulas. 4PM1 lists it as $f (g (x)) \to f^{'} (g (x)) g^{'} (x)$ with the formulas to learn.

Worked example

1
Find $\frac{d y}{d x}$ for $y = (3 x + 1)^{5}$ .
2
$u = 3 x + 1, \frac{d u}{d x} = 3$ — Name the inside and differentiate it.
3
$y = u^{5}, \frac{d y}{d u} = 5 u^{4}$ — Differentiate the outside with the power rule.
4
$\frac{d y}{d x} = 5 u^{4} \times 3$ — Multiply the two gradients.
5
$\frac{d y}{d x} = 15 (3 x + 1)^{4}$ — Put the inside back in place of u.

Key idea

Differentiate the outside and leave the inside alone. Then multiply by the derivative of the inside.

Powers and roots of a bracket

With practice you skip writing u. For any bracket raised to a power, use this shortcut:

Formula

\frac{d}{d x} [f (x)]^{n} = n [f (x)]^{n - 1} \times f^{'} (x)

$f (x)$ — the expression inside the bracket. It stays the same.
$n$ — the power. Any rational number, so roots and fractions work too.
$f^{'} (x)$ — the derivative of the inside. Never forget it.

Worked example

1
$y = 4 x^{2} + 9$ . Find $\frac{d y}{d x}$ and the gradient at $x = 2$ .
2
$y = (4 x^{2} + 9)^{\frac{1}{2}}$ — Write the root as the power one half.
3
$\frac{d y}{d x} = \frac{1}{2} (4 x^{2} + 9)^{- \frac{1}{2}} \times 8 x$ — Outside: bring down 1/2 and take 1 off. Inside: $4 x^{2} + 9$ gives 8x.
4
$\frac{d y}{d x} = \frac{4 x}{4 x ^{2} + 9}$ — Tidy: half of 8x is 4x, and the power −1/2 puts the root on the bottom.
5
$x = 2 : \frac{8}{16 + 9} = \frac{8}{5}$ — Substitute. 25 has an exact root, so no calculator is needed.

Worked example

1
$y = \frac{6}{( 2 x - 1 ) ^{3}}$ . Find $\frac{d y}{d x}$ .
2
$y = 6 (2 x - 1)^{- 3}$ — Bring the bracket to the top. Its power becomes negative.
3
$\frac{d y}{d x} = 6 \times (- 3) (2 x - 1)^{- 4} \times 2$ — Bring down −3, take 1 off to get −4, then times 2 for the inside.
4
$\frac{d y}{d x} = - \frac{36}{( 2 x - 1 ) ^{4}}$ — Multiply the numbers: 6 × (−3) × 2 = −36.

Common mistake

Forgetting to multiply by the inside's derivative. $\frac{d}{d x} (2 x - 1)^{4}$ is $8 (2 x - 1)^{3}$ , not $4 (2 x - 1)^{3}$ . Mark schemes give a separate mark for this factor.

Common mistake

Changing the inside. The bracket keeps $2 x - 1$ in the answer. Only the power changes.

Key idea

Power down, power minus one, bracket unchanged, times the inside's derivative.

e, sin and cos of an expression

The same idea works for every standard function. Use the standard rule on the outside, then multiply by the derivative of the inside.

Formula

\frac{d}{d x} (e^{f (x)}) = f^{'} (x) e^{f (x)}

$f (x)$ — the power. It stays the same; its derivative comes in front.

Formula

\frac{d}{d x} (sin f (x)) = f^{'} (x) cos f (x)

$f (x)$ — the angle, in radians. It stays the same inside the cos.

Formula

\frac{d}{d x} (cos f (x)) = - f^{'} (x) sin f (x)

$f (x)$ — the angle. Remember the minus from cos.

Worked example

1
$y = e^{x^{2} - 3 x}$ . Find the gradient at $x = 3$ .
2
$\frac{d y}{d x} = (2 x - 3) e^{x^{2} - 3 x}$ — The power stays. Its derivative, 2x − 3, comes in front.
3
$x = 3 : (6 - 3) e^{9 - 9} = 3 e^{0}$ — Substitute x = 3.
4
$= 3$ — e to the power 0 is 1.

A power of a trig function hides a chain too. $sin^{3} 2 x$ means $(sin 2 x)^{3}$ : the outside is the cube, and the inside is $sin 2 x$ .

Worked example

1
$y = sin^{3} 2 x$ . Find $\frac{d y}{d x}$ .
2
$y = (sin 2 x)^{3}$ — Rewrite so you can see the outside and the inside.
3
$\frac{d y}{d x} = 3 (sin 2 x)^{2} \times 2 cos 2 x$ — Outside: the power rule. Inside: sin 2x gives 2 cos 2x.
4
$\frac{d y}{d x} = 6 sin^{2} 2 x cos 2 x$ — Multiply the numbers and tidy.

Common mistake

Differentiating the angle and stopping. $\frac{d}{d x} sin (x^{2})$ is $2 x cos (x^{2})$ , not $cos 2 x$ . The angle inside stays $x^{2}$ .

Key idea

The inside always stays the same. Its derivative is the extra factor at the front.

ln and tan of an expression (0606/4037)

Rule

This step is for 0606 and 4037 only. 4PM1 does not ask you to differentiate ln or tan.

For ln, the outside rule is “1 over”. So the inside goes on the bottom and its derivative goes on top.

Formula

\frac{d}{d x} (ln f (x)) = \frac{f ^{'} ( x )}{f ( x )}

$f (x)$ — the expression inside the ln. It must be positive.
$f^{'} (x)$ — its derivative, on top.

Formula

\frac{d}{d x} (tan f (x)) = f^{'} (x) sec^{2} f (x)

$f (x)$ — the angle, in radians.

Worked example

1
$y = ln (x^{2} + 5)$ . Find the gradient at $x = 2$ .
2
$\frac{d y}{d x} = \frac{2 x}{x ^{2} + 5}$ — Inside on the bottom, its derivative on top.
3
$x = 2 : \frac{4}{4 + 5} = \frac{4}{9}$ — Substitute. Leave it as a fraction.

Worked example

1
$y = 2 tan 3 x + ln 4 x - 3$ . Find $\frac{d y}{d x}$ .
2
$2 tan 3 x \to 2 \times 3 sec^{2} 3 x = 6 sec^{2} 3 x$ — The inside, 3x, gives 3.
3
$ln 4 x - 3 = \frac{1}{2} ln (4 x - 3)$ — Log law first: the root is the power 1/2, and it comes to the front.
4
$\frac{1}{2} ln (4 x - 3) \to \frac{1}{2} \times \frac{4}{4 x - 3} = \frac{2}{4 x - 3}$ — Derivative of the inside over the inside.
5
$\frac{d y}{d x} = 6 sec^{2} 3 x + \frac{2}{4 x - 3}$ — Put the terms together.

Common mistake

Writing $\frac{1}{x ^{2} + 5}$ . That forgets the inside's derivative, 2x, on top.

Common mistake

Splitting a sum inside ln. $ln (x^{2} + 5)$ is not $ln x^{2} + ln 5$ . Log laws split products, not sums.

Key idea

ln of anything: derivative of the inside over the inside.

Using it: exact answers and dy/dx = 0

Exams often ask you to find $\frac{d y}{d x}$ , then use it: find an exact gradient, or solve $\frac{d y}{d x} = 0$ . Factorise the derivative, then ask which factor can be zero.

Worked example

1
$y = (2 + cos 2 x)^{3}$ . Solve $\frac{d y}{d x} = 0$ for $0 < x < π$ .
2
$\frac{d y}{d x} = 3 (2 + cos 2 x)^{2} \times (- 2 sin 2 x)$ — Outside: the power rule. Inside: cos 2x gives −2 sin 2x.
3
$\frac{d y}{d x} = - 6 sin 2 x (2 + cos 2 x)^{2}$ — Tidy into factors.
4
$2 + cos 2 x \geq 1$ — Cos is never below −1, so this bracket is never 0.
5
$sin 2 x = 0$ — So the only way to get 0 is from this factor.
6
$0 < 2 x < 2 π \Rightarrow 2 x = π$ — Double the range. Sin is 0 at 0, π and 2π, and only π is inside.
7
$x = \frac{π}{2}$ — Halve to get x.

Marks usually go: one for the outside part, one for the inside's derivative, then marks for solving. So write the unsimplified line first, then tidy.

Common mistake

Dividing by a factor without checking it. A factor like $e^{f (x)}$ or $2 + cos 2 x$ is never 0, so say so; a factor like $sin 2 x$ can be 0, so solve it.

Key idea

Find the derivative, factorise it, then set each factor that can be zero equal to zero.

Which method? Read the question

Spot the inside first. If there is no inside, you do not need the chain rule.

You see	You do
A bracket to a power, or a root of a bracket	Power rule on the bracket, like $(a x + b)^{n}$ , times the inside's derivative
$\frac{k}{( \dots ) ^{n}}$	Bring it up as a negative power first
$e^{f (x)}$	$f^{'} (x) e^{f (x)}$
$sin f (x)$ or $cos f (x)$	$f^{'} (x) cos f (x)$ or $- f^{'} (x) sin f (x)$
$sin^{n} x$	Rewrite as $(sin x)^{n}$ ; the outside is the power
$ln f (x)$ (0606 / 4037)	$\frac{f ^{'} ( x )}{f ( x )}$ ; use log laws first
$tan f (x)$ (0606 / 4037)	$f^{'} (x) sec^{2} f (x)$

Key idea

Today's AI systems learn with the chain rule. To train a neural network, a computer works out how each setting affects the final error, link by link through the network. This method, called backpropagation, is the chain rule used millions of times.

Common mistake

Using the chain rule on two functions multiplied together, like $x^{2} e^{x}$ . That is a product: see the product rule lesson.

Key idea

Outside first, inside unchanged, then times the derivative of the inside.

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