8 min readPathway sheet1-mark MCQ; 2-mark questions; used inside almost every derivative in the paper

Chain rule: derivative of a composite function

For y = f(g(x)), differentiate the outer function first, then multiply by the derivative of the inner one: dy/dx = f′(g(x)) × g′(x).

Chain ruleIf y = f(t) and t = g(x), then dy/dx = (dy/dt) × (dt/dx)

Three layersIf y = f(u), u = g(v) and v = h(x), then dy/dx = (dy/du) × (du/dv) × (dv/dx)

Power of a functiond/dx [g(x)]ⁿ = n[g(x)]ⁿ⁻¹ × g′(x)

The idea

A composite function is a function inside another, like sin(x²): first square x, then take the sine. To differentiate it, work from the outside in. Differentiate the outer function, keeping the inside unchanged. Then multiply by the derivative of the inside. Repeat for every layer.

How to use the chain rule

Worked on y = tan(√(x² + 1)).

1
Spot the layers: tan, then √, then x² + 1.
2
Differentiate the outer layer, leaving the inside as it is: sec²(√(x² + 1)).
3
Multiply by the derivative of the next layer: 1/(2√(x² + 1)).
4★
Multiply by the derivative of the innermost layer: 2x.
5
Simplify: dy/dx = x sec²(√(x² + 1))/√(x² + 1).

★ key step

Example 1: two layers

Problem

Differentiate (i) y = sin(x² + 3x), (ii) y = (3x² − 5x + 2)⁴.

Working

(i) Outer: sin, whose derivative is cos. Inner: x² + 3x, whose derivative is 2x + 3.
dy/dx = cos(x² + 3x) × (2x + 3)
(ii) Outer: a fourth power, whose derivative is 4( )³. Inner: 3x² − 5x + 2, whose derivative is 6x − 5.
dy/dx = 4(3x² − 5x + 2)³ × (6x − 5)

Answer

(i) (2x + 3) cos(x² + 3x); (ii) 4(6x − 5)(3x² − 5x + 2)³

Example 2: three layers

Problem

Differentiate y = cos²(3x).

Working

Write y = [cos(3x)]². The layers are: square, cos, 3x.
Square: 2 cos(3x)
cos: multiply by −sin(3x)
3x: multiply by 3
dy/dx = 2 cos 3x × (−sin 3x) × 3 = −6 sin 3x cos 3x
Using 2 sin θ cos θ = sin 2θ: dy/dx = −3 sin 6x

Answer

dy/dx = −3 sin 6x

The common mistake

Wrongd/dx sin(5x) = cos(5x). The derivative of the inside, 5, is missing.

Rightd/dx sin(5x) = cos(5x) × 5 = 5 cos 5x.

After every step, ask: have I multiplied by the derivative of what is inside?

In real life

The level of a medicine in the blood depends on time, and its effect depends on that level. The chain rule links them: rate of change of effect = (effect per unit level) × (level per unit time).

Exam angle

1 marks: “If y = sin³x, find dy/dx.” Outer: cube; inner: sin x. dy/dx = 3 sin²x × cos x = 3 sin²x cos x.

2 marks: “Differentiate y = √(sin 2x) with respect to x.” dy/dx = [1/(2√(sin 2x))] × cos 2x × 2 = cos 2x/√(sin 2x).

2 marks: “Differentiate y = sin(x²) cos(x³).” Product rule with the chain rule: dy/dx = 2x cos(x²) cos(x³) − 3x² sin(x²) sin(x³).

Remember it

Outside in, and multiply as you go.