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).
After dividing by hn x^(n-1) + (terms with h that vanish as h→0).
Resultd/dx(xⁿ) = n x^(n-1).
LinearityDerivative of sum is sum of derivatives — provable from limit definition.
Constantd/dx(c) = lim (c−c)/h = 0.
Setupf(x)=sin x: [sin(x+h)−sin x]/h.
Use identitysin(x+h) = sin x cos h + cos x sin h.
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).
What is the difference between simple and compound interest?✗ Simple and compound interest give the same total — compound just uses a different formula. ✓ Simple interest is calculated only on the original amount; compound interest is calculated on the original amount plus accumulated interest.
What is the difference between the expression 3x + 5 and the equation 3x + 5 = 11?✗ Expressions and equations are the same — you can solve either one. ✓ An expression has no equals sign and cannot be solved; an equation states two things are equal and can be solved for x.
On a speed-time graph, what does the area under the line represent?✗ The total speed achieved. ✓ The distance travelled.
Practise Calculus — free games
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