A-Level Maths: Calculus Revision

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What is the derivative of a constant, like d/dx(5)?

Calculus is a core part of A-Level Maths. Revise the key concepts and common mistakes below, then lock them in with the free games.

Key concepts

DerivativeRate of change of f at x: f'(x) = limₕ→₀ [f(x+h) − f(x)]/h. Geometric meaning: gradient of tangent.

Power ruled/dx(xⁿ) = nxⁿ⁻¹ for any real n. Linear: d/dx(c f(x)) = c f'(x).

Standard derivativesd/dx(sin x) = cos x; d/dx(cos x) = −sin x; d/dx(eˣ) = eˣ; d/dx(ln x) = 1/x.

Stationary pointsPoints where f'(x) = 0. Use second derivative or sign change to classify max, min, or point of inflection.

Second derivativef''(x) = d²y/dx². f''(x) > 0 indicates concave up (minimum); f''(x) < 0 indicates concave down (maximum).

Rates of changeIf y = f(x), then dy/dt = (dy/dx)(dx/dt). Used in related-rates problems (e.g., expanding circles, draining tanks).

General xⁿExpand (x+h)ⁿ using binomial theorem; subtract xⁿ; divide by h; take limit.

Binomial expansion(x+h)ⁿ = xⁿ + n x^(n-1) h + O(h²).

Differentiating xⁿ from first principlesAfter dividing by h you get nxⁿ⁻¹ plus terms in h, which vanish as h → 0.

Power ruled/dx(xⁿ) = nxⁿ⁻¹.

LinearityDerivative of sum is sum of derivatives — provable from limit definition.

Constantd/dx(c) = lim (c−c)/h = 0.

Differentiating sin x from first principlesf′(x) = lim (h→0) [sin(x+h) − sin x] / h.

Compound angle identitysin(x+h) = sin x cos h + cos x sin h, used to differentiate sin x from first principles.

What your exam board covers

Where Calculus sits in each exam board's specification. Tap a board to see what it lists.

AQADifferentiationSpec 7357

Part of: Pure mathematics

Source: AQA A-level Mathematics (7357) specification — Subject content, accessed 23 June 2026.

Learn Calculus step by step

Free interactive mini-lessons: short screens, worked examples and quick checks. Pick your exam board:

AQADifferentiation

Other boards:CCEA Edexcel OCR

Common mistakes to avoid

Questions where students often pick the tempting wrong answer — make sure you know the right one:

What is the derivative of a constant, like d/dx(5)?✗ The derivative of 5 is 5 because the constant stays the same. ✓ 0 — constants do not change with x.

What is the derivative of (3x + 1)^2 with respect to x?✗ The derivative is 2(3x + 1) — applying the power rule without the chain rule. ✓ 6(3x + 1) — by the chain rule, multiplying by the derivative of the inside (3).

Practise Calculus — free games

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Calculus

Key concepts

What your exam board covers

Common mistakes to avoid

Practise Calculus — free games

More A-Level Maths topics