A-level · Beta

The chain rule

Separate inner and outer functions and multiply their rates of change.

Change the inside function u = ax + b. Compare its rate with the outer rate.

y=u^{3},\quad u=2x+(1),\quad\frac{dy}{dx}=3u^{2}\times2

At this point: 3 × 2 = 6.

The outside rate alone is incomplete: multiply by the inside rate. Sine inputs are in radians. Displayed values are rounded.

Predict, test and explain

Step 1 of 3

Predict

What is the derivative of (2x + 1)³?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

What is the derivative of (2x + 1)³?

$y=(2x+1)^3$

Hint

Differentiate the outside, keeping the inside intact; multiply by the inside derivative.

Reveal answer

Multiply the outer derivative 3u² by the inner derivative 2.

$6(2x+1)^2$

Hint 1

Differentiate the outside, keeping the inside intact; multiply by the inside derivative.

Hint 2

Multiply the outer derivative 3u² by the inner derivative 2.

Hint 3

Compare the result with the model, then explain why it occurs.

Worked example

This example uses fixed values, separate from the diagram controls.

Differentiate y = sin(3x).

Answer: dy/dx = 3cos(3x), with x in radians.

Connect this idea

Watch out: Remember the inside derivative; the outer derivative alone is incomplete.

Specification and learning route

AQA G4 · Edexcel Pure 7.4

A-level extension. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.

Useful starting knowledge: Basic derivatives; Composite functions; Algebra.