Overview
Differentiation is used to find maximum/minimum points, tangents, and rates of change.
Stationary points
At a stationary point dy/dx = 0. To find its nature use the second derivative:
- d²y/dx² > 0 → minimum
- d²y/dx² < 0 → maximum
(Or check the sign of the gradient either side.)
Tangents and normals
- The tangent has gradient dy/dx at that point.
- The normal is perpendicular, so its gradient is −1 ÷ (dy/dx).
Rates of change
If two quantities are connected, link their rates with the chain rule: dA/dt = dA/dr × dr/dt.
Worked example
y = x² − 4x + 1: dy/dx = 2x − 4 = 0 → x = 2. d²y/dx² = 2 (> 0) → minimum at (2, −3).
Common mistakes
- Stopping at dy/dx = 0 without testing whether it's a max or min.
Exam tips
- Optimisation: form an equation, differentiate, set = 0, then check it's the
max/min asked for.