Chain rule | Year 12 | Ascent Library

01 / LEARN

The idea, step by step

Track the inner change

If $y = F (g (x))$ , then $d y / d x = F^{'} (g (x)) g^{'} (x)$ . Differentiate the outer operation at the unchanged inner expression, then multiply by the inner derivative.

A useful annotation is $u = g (x)$ , followed by $d y / d u$ and $d u / d x$ . It reveals a missing factor before algebra hides it.

Combining rules

A product or quotient may contain composite factors. First choose the rule for the whole expression, then use the chain rule inside the required derivatives.

Check real domains, especially for roots and negative powers. A square root may be defined at an endpoint even when its derivative is not finite there.

A common trap

Reducing the outer power without multiplying by the inner derivative.

Instead: The inner quantity changes at its own rate; multiply by that rate.

02 / WORKED EXAMPLES

See the thinking

Example 1

A linear inner function

Differentiate $y = {(3 x - 1)}^{5}$ .

The outer derivative is $5 u^{4}$ and the inner derivative is 3.
Multiply and replace $u$ with $3 x - 1$ .

15 {(3 x - 1)}^{4}

Example 2

A root with a quadratic input

Differentiate $y = \sqrt{x^{2} + 4}$ .

Write $y = {(x^{2} + 4)}^{1 / 2}$ .
The chain rule gives $\frac{1}{2} {(x^{2} + 4)}^{- 1 / 2} (2 x)$ .

\frac{x}{\sqrt{x^{2} + 4}}

Check your understanding

Differentiate ${(2 x + 5)}^{- 2}$ .

Show answer

$- 4 {(2 x + 5)}^{- 3}$ , $x \neq - 5 / 2$

The outer derivative contributes $- 2$ and the inner derivative contributes 2.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Differentiate ${(4 x + 3)}^{6}$ .
2 marks
Differentiate ${(x^{2} - 2 x)}^{3}$ .
2 marks
Differentiate $1 / (x^{2} + 1)$ .
2 marks
Differentiate $x {(x + 1)}^{4}$ .
3 marks
Find the gradient of $y = {(5 - x)}^{3}$ at $x = 2$ .
2 marks
Find the derivative of $y = \sqrt{2 x - 1}$ and state its domain.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Differentiate ${(4 x + 3)}^{6}$ .
2 marks
The outer derivative is $6 {(4 x + 3)}^{5}$ .
Multiply by 4.
$24 {(4 x + 3)}^{5}$
Differentiate ${(x^{2} - 2 x)}^{3}$ .
2 marks
Differentiate the cube.
Multiply by the derivative of $x^{2} - 2 x$ .
$3 {(x^{2} - 2 x)}^{2} (2 x - 2)$
Differentiate $1 / (x^{2} + 1)$ .
2 marks
Write the function as ${(x^{2} + 1)}^{- 1}$ .
Multiply the outer derivative by $2 x$ .
$- 2 x / {(x^{2} + 1)}^{2}$
Differentiate $x {(x + 1)}^{4}$ .
3 marks
Use the product rule to obtain ${(x + 1)}^{4} + 4 x {(x + 1)}^{3}$ .
Factor and simplify.
${(x + 1)}^{3} (5 x + 1)$
Find the gradient of $y = {(5 - x)}^{3}$ at $x = 2$ .
2 marks
The derivative is $- 3 {(5 - x)}^{2}$ .
Substitute 2.
$- 27$
Find the derivative of $y = \sqrt{2 x - 1}$ and state its domain.
3 marks
The outer derivative is one half times power $- 1 / 2$ .
Multiply by 2; a finite derivative requires $2 x - 1 > 0$ .
$1 / \sqrt{2 x - 1}$ for $x > 1 / 2$

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Natural exponential function and its derivative

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Curriculum reference & lesson scope

SACE Stage 2 Mathematical Methods

Further differentiation and applications from the current SACE Methods outline. Logarithmic derivatives have their own chapter; integration follows later. Trigonometric calculus uses radians.

Official curriculum source ↗

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