Differentiation

Differentiation finds the gradient of a curve at any point — the rate at which $y$ changes with $x$. It unlocks tangents, normals and turning points.

The power rule

$$\text{if } y = x^n \text{ then } \frac{dy}{dx} = n x^{n-1}$$

Multiply by the power, then reduce the power by one. First rewrite roots and fractions as powers: $\sqrt{x} = x^{1/2}$, $\dfrac{1}{x^2} = x^{-2}$

Worked example. $y = 4x^3 - \dfrac{2}{x} = 4x^3 - 2x^{-1}$