Tangents, normals and stationary points
The derivative gives the gradient of a curve, so it is used to find tangents and normals and to locate turning points. A stationary point occurs where dy⁄dx = 0.
Key idea
At a stationary point dy⁄dx = 0. Use the second derivative to classify it: if d2y⁄dx2 > 0 it is a minimum; if d2y⁄dx2 < 0 it is a maximum.
Increasing, decreasing and rates of change
A function is increasing where dy⁄dx > 0 and decreasing where dy⁄dx < 0. The chain rule links connected rates of change; for example, dA⁄dt = (dA⁄dr)(dr⁄dt).
Worked example
For y = x2 − 6x + 5, dy⁄dx = 2x − 6 = 0 gives x = 3. Since d2y⁄dx2 = 2 > 0 this is a minimum. The minimum value is y = 9 − 18 + 5 = −4.
Remember
- Set dy⁄dx = 0 to find stationary points, then test the nature.
- The normal gradient is −1⁄(tangent gradient).
- For connected rates, build a chain of derivatives.