Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Differentiation

In one sentence. The gradient of a curve at a single point — the power rule, and why it works. Higher Tier, and genuinely on this specification.

Calculus — one of 2 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Higher Tier only.

This module ends with Isaac Newton. Learn Differentiation, teach it, then defend it against Isaac Newton.

Where this sits in the specification

Specification 4MA1, 3.4 Calculus (Higher). Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 3.4 — Calculus (Higher)

1 · Understand it

What it is

A curve's steepness changes from point to point — differentiation gives a formula for the gradient at ANY point. The rule: for y = axⁿ, the derivative is dy/dx = naxⁿ⁻¹ ('multiply by the power, drop the power by one'). Constants vanish; terms differentiate independently.

Why it works

Zoom in on any smooth curve and it straightens: the gradient at a point is the gradient of that zoomed-in line. For y = x², the gradient between x and a nearby x + h works out to 2x + h; as the zoom tightens (h shrinks to nothing) it settles at exactly 2x. The power rule is that zooming argument done once for every power. Constants vanish because adding 7 lifts the whole curve without tilting it anywhere.

When a question is asking for it

'Find dy/dx', 'find the gradient of the curve at x = …' (differentiate, THEN substitute), and as the engine for turning points and kinematics in the next module.

How to hold it in your head

dy/dx is a gradient-o-meter: a new formula that reports steepness wherever you point it. y = x² has meter 2x — at x = 3 the curve climbs at rate 6, at x = 0 it's momentarily flat, at x = −3 it falls at rate 6. One formula, every point.

2 · The mistakes that cost marks

  • Differentiating axⁿ as axⁿ⁻¹ (forgetting to multiply by the power): x³ → 3x², not x².
  • Keeping constants: the derivative of +7 is 0, not 7. Flat additions never tilt the curve.
  • Substituting BEFORE differentiating: the gradient at x = 4 means differentiate first, then plug in 4.
  • Treating dy/dx as a fraction to cancel. It's one symbol: 'the derivative of y with respect to x'.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

y = 4x³ − 6x² + 7. Find dy/dx.

  1. 4x³ → 12x² (multiply by 3, drop to power 2).
  2. −6x² → −12x.
  3. +7 → 0.

Answer: dy/dx = 12x² − 12x

Find the gradient of y = x² − 3x at the point where x = 4.

  1. Differentiate: dy/dx = 2x − 3.
  2. Substitute x = 4: 2(4) − 3 = 5.

Answer: 5

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — y = x³. Find dy/dx. (Type it like 3x^2.)
  2. Grade 7 — y = 4x³ − 6x² + 7. Find dy/dx. (Type it like 12x^2-12x.)
  3. Grade 8 — Work out the gradient of the curve y = x² − 3x at the point where x = 4.
  4. Grade 9 — Find the coordinates of the turning point of y = x² − 6x + 11. (Type it like (3,2).)
  5. 9+ — y = 2x³ − 9x² + 12x. The curve has two turning points. Work out the x-coordinate of the maximum point.

Ask the person who made it

Inside Incandio you can put a question to Isaac Newton, who answers knowing exactly which module you have just finished: “You invented calculus because the mathematics of your time couldn't handle things that CHANGE — falling apples, orbiting moons. What was the problem that forced you to invent a whole new mathematics, and what was the key idea that cracked it?”

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Differentiation, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Differentiation works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Differentiation against Isaac Newton — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 10 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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