Differentiation

The derivative as a gradient, differentiating powers of x, and finding tangents, normals and stationary points.

9 min read

The derivative $\frac{d y}{d x}$ gives the gradient of the tangent to a curve, the instantaneous rate of change. Differentiation lets you find tangents, normals and turning points.

For $y = x^{n}$ (any rational $n$ ), $\frac{d y}{d x} = n x^{n - 1}$ . Differentiate sums term by term; constants differentiate to $0$ . Notation: $\frac{d y}{d x}$ , $f^{'} (x)$ , and the second derivative $\frac{d ^{2} y}{d x ^{2}}$ , $f^{''} (x)$ .

Worked example

Differentiate $y = 3 x^{4} - \frac{2}{x} + 5$ .

Rewrite $\frac{2}{x} = 2 x^{- 1}$ : $\frac{d y}{d x} = 12 x^{3} + 2 x^{- 2} = 12 x^{3} + \frac{2}{x ^{2}} .$

A curve with the tangent line and the perpendicular normal line drawn at a point, and a stationary point marked at the bottom. — The derivative gives the tangent's gradient; the normal is perpendicular to it. At a stationary point the gradient is zero.

The gradient at a point is $\frac{d y}{d x}$ evaluated there. The normal is perpendicular to the tangent, so its gradient is $- \frac{1}{tangent gradient}$ .

Worked example

Find the tangent to $y = x^{2}$ at $x = 3$ .

$\frac{d y}{d x} = 2 x = 6$ at $x = 3$ ; the point is $(3, 9)$ . Tangent: $y - 9 = 6 (x - 3)$ , i.e. $y = 6 x - 9.$

At a stationary point $\frac{d y}{d x} = 0$ . Classify with the second derivative: $\frac{d ^{2} y}{d x ^{2}} > 0$ gives a minimum, $< 0$ gives a maximum. A function is increasing where $f^{'} (x) > 0$ and decreasing where $f^{'} (x) < 0$ .

Worked example

Find and classify the stationary points of $y = x^{3} - 3 x$ .

$\frac{d y}{d x} = 3 x^{2} - 3 = 0 \Rightarrow x = \pm 1$ . Since $\frac{d ^{2} y}{d x ^{2}} = 6 x$ : at $x = 1$ it is $+ 6$ (minimum), at $x = - 1$ it is $- 6$ (maximum).

Forgetting to rewrite terms like $\frac{1}{x}$ or $x$ as powers before differentiating.
Using the tangent gradient for the normal (it should be the negative reciprocal).
Mixing up the max/min test: $f^{''} > 0$ is a minimum.

Differentiation

Overview

Differentiating powers

Tangents & normals

Stationary points & increasing/decreasing

Common mistakes