Chain rule | Year 12 | Ascent Library

01 / LEARN

Understand the idea.

Track the inner change

A composite function applies one operation to the result of another. Set $u = g (x)$ : the chain rule is $\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x}$ . The outer rate must be multiplied by the inner rate.

Differentiate the outer operation while leaving its input intact, then differentiate that input. For a power, this means lowering the outer power without first changing the expression inside its brackets.

Combining rules

Choose the rule for the entire expression first. In a product, apply the product rule; then use the chain rule whenever one of its factors contains a function inside another function.

State where both the function and derivative are defined. Roots and negative powers need particular care: an endpoint may belong to a function's domain even though its tangent gradient is not finite there.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A linear inner function

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

Worked solution

1
Name the inner function.
$u = 3 x - 1, y = u^{5}$
The fifth power is applied to the entire linear expression.
2
Find the two rates.
$\frac{d y}{d u} = 5 u^{4}, \frac{d u}{d x} = 3$
The power rule applies to u; the inner function has constant gradient 3.
3
Multiply and restore x.
$\frac{d y}{d x} = 5 {(3 x - 1)}^{4} \cdot 3 = 15 {(3 x - 1)}^{4}$
The chain rule accounts for the faster change of the inner input.

Answer

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

EXAMPLE 02

A root with a quadratic input

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

Worked solution

1
Write the root as a power and check its input.
$y = {(x^{2} + 4)}^{1 / 2}, x^{2} + 4 \geq 4 > 0$
The function and its root derivative are defined for every real x.
2
Apply the outer power rule and inner derivative.
$\frac{d y}{d x} = \frac{1}{2} {(x^{2} + 4)}^{- 1 / 2} (2 x)$
The derivative of the inner quadratic is 2x.
3
Simplify using a positive square root.
$\frac{d y}{d x} = \frac{x}{\sqrt{x^{2} + 4}}, x \in ℝ$
The denominator never vanishes, so no additional inputs are excluded.

Answer

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

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

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

The outer derivative is $- 2 {(2 x + 5)}^{- 3}$ and the inner derivative is $2$ . Their product is $- 4 {(2 x + 5)}^{- 3}$ ; exclude $x = - \frac{5}{2}$ because the original negative power is undefined there.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Differentiate ${(4 x + 3)}^{6}$ .
2 marks
Differentiate ${(x^{2} - 2 x)}^{3}$ .
2 marks
Differentiate $\frac{1}{x^{2} + 1}$ .
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 $\frac{1}{x^{2} + 1}$ .
2 marks
Write the function as ${(x^{2} + 1)}^{- 1}$ .
Multiply the outer derivative by $2 x$ .
$- \frac{2 x}{{(x^{2} + 1)}^{2}}$
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 $- \frac{1}{2}$ .
Multiply by 2; a finite derivative requires $2 x - 1 > 0$ .
$\frac{1}{\sqrt{2 x - 1}}$ for $x > \frac{1}{2}$

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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 ↗

In this lesson

Understand the idea.

Track the inner change

Combining rules

See the method in action.

A linear inner function

Name the inner function.

Find the two rates.

Multiply and restore x.

A root with a quadratic input

Write the root as a power and check its input.

Apply the outer power rule and inner derivative.

Simplify using a positive square root.

Try it yourself.

Check your understanding

Keep practising