Differentiation (AS Unit 1: Pure Mathematics A) — WJEC A-Level Mathematics
Test yourself on Differentiation (AS Unit 1: Pure Mathematics A) with WJEC A-Level practice questions.
7 days Premium · Then free forever · No card, no charge
Differentiation (AS Unit 1: Pure Mathematics A) explained
The second derivative d²y/dx² (or f''(x)) is obtained by differentiating the first derivative dy/dx with respect to x.
Read the full explanation
Geometrically, it measures the rate at which the gradient of the curve is changing. At a stationary point where dy/dx = 0, the second derivative classifies the point: if d²y/dx² > 0, the gradient is increasing through zero from negative to positive, meaning the curve bends upwards to form a local minimum; if d²y/dx² < 0, the gradient is decreasing from positive to negative, bending downwards to form a local maximum. If d²y/dx² = 0, the test is inconclusive, requiring examination of the sign of dy/dx on either side of the point to determine whether it is a local extremum or a point of inflection.
Your focus
- Calculate the second derivative d²y/dx² for algebraic and polynomial functions.
- Apply the second derivative test to classify stationary points as local maxima or minima.
- Interpret the second derivative geometrically as the rate of change of gradient.
Differentiation (AS Unit 1: Pure Mathematics A) exam tips
Marking Points
- differentiating the first derivative correctly to calculate d²y/dx²
- substituting stationary point x-values into d²y/dx² to evaluate its sign
- concluding that d²y/dx² > 0 indicates a local minimum and d²y/dx² < 0 indicates a local maximum
Examiner Tips
- 💡State your reasoning explicitly: 'dy/dx = 0 and d²y/dx² > 0, therefore local minimum'.
- 💡If d²y/dx² = 0, test the gradient just to the left and right of the stationary point.
Common Mistakes
- reversing the second derivative criteria, claiming d²y/dx² > 0 gives a maximum
- assuming d²y/dx² = 0 automatically implies a point of inflection without testing the sign of dy/dx