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The second derivative d²y/dx² measures how a gradient is changing and classifies stationary points as maxima, minima or points of inflection. This topic covers finding stationary points, testing their nature and applying this to optimisation. It appears across A-Level and iAL Pure Maths with the major exam boards. Learn it free on eClassroom through guided worked examples and instantly-marked exam-style questions.
How you do it
Set the first derivative to zero and solve for the stationary points, then substitute back for the y values. Differentiate again: a positive second derivative means a minimum, negative means a maximum. If it is zero, test gradients either side instead.
Before you start
It assumes you can handle basic differentiation. You meet it in Pure 2. From here the path runs to classifying turning points and optimisation. The topic sits within Differentiation.
What you learn
At a stationary point a curve flattens out — its gradient is zero. The second derivative then tells you whether it’s a peak or a valley. A stationary point is where the gradient is zero. Positive second derivative → minimum (valley); negative → maximum (peak). The course works through finding stationary points, the second derivative, maximum or minimum?, when the second derivative is zero and optimisation. Every section has instant-feedback practice questions matched to Edexcel International A-Level and 9MA0 Pure, and the course finishes with full exam-style questions you mark yourself.
Related Topics
We teach Second Derivatives & Stationary Points as one of the Differentiation topics in our A-level course library, at the AS year. The course is built from 5 concept sections, 5 worked examples and 5 check points before the exam-style set. It sits alongside Standard Functions, Tangents & Normals, Basic Differentiation and Chain Rule in the same strand. Endless extra practice on this strand is available on the Differentiation revision engine, which generates a fresh question each time.
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