Differentiation
A-level Maths (9MA0) · exam-style practice, examiner-report intelligence and the tools that drill it.
The topic on one screen
- First principles is on the spec and keeps appearing: f'(x) = lim(h → 0) of [f(x + h) - f(x)] / h. Learn the structure, not just the answer.
- Differentiation from first principles for sin x and cos x uses the compound-angle formula plus the small-h limits — both are given in the question when needed.
- Product, quotient and chain rules: most marks die on sloppy bracketing, not on the rule itself.
- Tangents and normals: gradient at the point first, then y - y1 = m(x - x1). The normal gradient is -1/m.
- Connected rates of change: write down dV/dt = dV/dr x dr/dt (chain rule) BEFORE substituting anything.
- Increasing/decreasing and stationary points: set f'(x) = 0, then justify the nature with f'' or a sign check.
Where students actually lose marks
In first-principles proofs the limit structure carries the marks: the mark scheme requires the full lim(h → 0) statement with correct fraction, and a conclusion — jumping straight to the answer scores the method nothing.
June 2024 Paper 1 mark scheme (Q04)
Where a question says 'solutions relying entirely on calculator technology are not acceptable', the working IS the answer — mark schemes award cso (correct solution only) marks that die on any unshown step.
June 2023 Paper 1 question paper (Q02, Q04 instruction)
In rates-of-change modelling the mark scheme explicitly refuses invented constants: you 'cannot just make up a value for k' — the constant must come from the given information.
June 2024 Paper 1 mark scheme (rates model)
Try it — exam-style
Given y = x2, use differentiation from first principles to show that dy/dx = 2x.
The curve y = x3 - 4x + 1 passes through the point (2, 1). Find the equation of the tangent to the curve at that point.
Differentiate y = (3x2 + 1)5 with respect to x.
A spherical balloon is inflated so its volume increases at 100 cm3 s−1. Find the rate of increase of the radius when r = 5 cm. (V = (4/3)πr3)
Questions are written in the style of past Edexcel papers (source shown on each) — never copied from them.
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Stuck on differentiation?
First-principles proofs are pure method marks — I drill the exact limit structure the mark scheme wants, and your first lesson is free.