Pure Maths
A-Level Differentiation
Differentiation measures rate of change. It gives gradients, tangents, normals, stationary points and the shape of a curve.
What you need to know
- Differentiating powers of x, including negative and fractional powers
- Chain, product and quotient rules
- Stationary points and the second derivative test
- Tangents, normals and connected rates of change
Key concepts
- At a stationary point dy/dx = 0; the sign of d²y/dx² classifies it.
- The normal gradient is the negative reciprocal of the tangent gradient.
- Rewrite roots and fractions as powers of x before differentiating.
Common mistakes
- Forgetting the inner derivative in the chain rule
- Sign slips when differentiating negative powers
- Stopping at dy/dx = 0 without classifying the point
Example question
Differentiate .
Show explanation
Multiply by the power and reduce it by one: 3 × 4x³ = 12x³.
Practise Differentiation
4 original questions with worked explanations, mixed across easy, medium and challenging difficulty.
Differentiation questions students ask
- When do I need the product rule instead of expanding?
- If expanding is quick, expand. The product rule matters when a factor is not a simple polynomial.