The gradient of a straight line is the same everywhere, but a curve keeps bending, so its steepness changes from point to point. Differentiation is the machine that gives a formula — written dy/dx — for the gradient at any point on a curve. It is the gateway to calculus and one of the most powerful tools in the whole syllabus.
1. The rule for powers of x
For a term axn, the derivative is anxn−1: multiply by the power, then knock the power down by one. Differentiate each term separately; a constant on its own differentiates to 0.
| Term y | dy/dx | Reason |
|---|---|---|
| x³ | 3x² | bring down 3, power → 2 |
| 5x² | 10x | 5 × 2 = 10, power → 1 |
| 4x | 4 | 4x¹ → 4x⁰ = 4 |
| 7 | 0 | a constant has no gradient |
2. Gradient at a point, and the tangent
- To get the gradient at x = a, substitute a into dy/dx.
- The tangent is the straight line just touching the curve there. Its gradient m equals dy/dx at that point; then use y − y₁ = m(x − x₁) through the point.
3. Stationary (turning) points: dy/dx = 0
At a peak or a valley the curve is momentarily flat, so its gradient is zero. Solve dy/dx = 0 to find the x-coordinate(s), then substitute back for y. To classify:
- Maximum — gradient goes + then 0 then −; the second derivative d²y/dx² < 0.
- Minimum — gradient goes − then 0 then +; d²y/dx² > 0.
Exam technique: the command word Find the equation of the tangent expects three shown steps — differentiate, evaluate the gradient, use y − y₁ = m(x − x₁). Each is a mark.