A-level maths / Pure 1 / Differentiation
Differentiation
Find a gradient, explain a change and choose the best possible dimensions.
Start with first principles to understand the idea, or choose the method you need. Move the graphs yourself, keep the algebra in view and check your working with hints and full solutions.
- Differentiation from first principlesFind a curve’s gradient from nearby points. Understand secants, tangents, the difference quotient and limits, with first-principles proofs and worked practice.
- Differentiation: the power ruleDifferentiate powers, polynomials, roots and reciprocals. Rewrite products and fractions, find second derivatives and solve gradient and parameter conditions.
- Tangents and normalsFind tangent and normal equations using derivatives. Handle horizontal tangents, intersections, external-point tangents and second meetings with a curve.
- Derivative graphs and increasing or decreasing functionsRead a derivative graph, find increasing and decreasing intervals, and connect slope, stationary points and concavity without confusing height with gradient.
- Stationary points: maxima, minima and inflectionFind stationary-point coordinates and classify them using sign changes or second derivatives. Handle inconclusive tests, domain restrictions, local extrema and ranges.
- Rates of changeInterpret derivatives with units and distinguish average from instantaneous rates. Apply differentiation to volume, displacement, velocity, acceleration and physical graphs.
- Differentiation and optimisation problemsUse differentiation to maximise area or volume and minimise distance or material. Build a one-variable model, apply physical domains and justify the optimum.
Useful foundations
Review index laws, straight-line graphs and algebraic methods when the preparation needs a refresher.