Differentiation — Finding the Gradient

Learn differentiation from scratch — what it means, the power rule, finding gradients, stationary points, and real-world applications. A-Level maths made accessible.

Differentiation sounds terrifying. But it's really just a way of finding the gradient (steepness) of a curve at any point. A straight line has one gradient everywhere. A curve's gradient changes — differentiation lets you calculate it.

The Power Rule

This is the rule you'll use 90% of the time:

If y = xⁿ then dy/dx = nxⁿ⁻¹

In English: bring the power down to the front, then reduce the power by 1.

y =	dy/dx =	How?
x³	3x²	Bring 3 down, reduce to 2
5x⁴	20x³	5 × 4 = 20, reduce power
x²	2x	Bring 2 down, x¹ = x
3x	3	x¹ → 1×x⁰ = 1
7 (constant)	0	Constants vanish

Worked Example 1 — Differentiating a Polynomial

Find dy/dx for y = 3x⁴ − 2x² + 5x − 1

3x⁴ → 12x³
−2x² → −4x
5x → 5
−1 → 0
dy/dx = 12x³ − 4x + 5

Finding the Gradient at a Specific Point

Worked Example 2 — Gradient at a Point

y = x³ − 3x + 2. Find the gradient when x = 2.

dy/dx = 3x² − 3
Substitute x = 2: 3(4) − 3 = 12 − 3 = 9

The gradient at x = 2 is 9 — the curve is climbing steeply upward.

Stationary Points (Maxima & Minima)

At the top of a hill or bottom of a valley, the gradient is zero. Set dy/dx = 0 to find these points.

Worked Example 3 — Finding Stationary Points

y = x² − 6x + 10. Find the stationary point and determine if it's a max or min.

dy/dx = 2x − 6
Set = 0: 2x − 6 = 0 → x = 3
y-value: 9 − 18 + 10 = 1. Point is (3, 1).
Second derivative: d²y/dx² = 2 (positive → minimum)

Real-World Connection

If distance is a function of time, its derivative is speed. If speed is a function of time, its derivative is acceleration. Differentiation measures rate of change — how fast something is changing at any instant.

Try It Yourself

Differentiate y = 4x³ + 2x − 7. (Answer: 12x² + 2)
Find the gradient of y = x² + 3x at x = −1. (Answer: −2 + 3 = 1)
Find the stationary point of y = x² − 4x + 5. (Answer: (2, 1), minimum)