Tangents & Normals

This lesson covers finding the equations of tangents and normals to curves — a fundamental application of differentiation in A-Level Mathematics. These ideas connect algebra, coordinate geometry, and calculus, and appear frequently on AQA papers.

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Tangent Lines

A tangent to a curve at a point P is the straight line that touches the curve at P and has the same gradient as the curve at that point.

To find the equation of a tangent to y = f(x) at x = a:

1. Find y = f(a) — the y-coordinate of the point.
2. Find the gradient: m = f'(a).
3. Use the point-gradient form: y − y₁ = m(x − x₁).

Example 1

Find the equation of the tangent to y = x³ − 2x + 1 at the point where x = 2.

Step 1: y = 8 − 4 + 1 = 5. So the point is (2, 5).

Step 2: dy/dx = 3x² − 2. At x = 2: m = 3(4) − 2 = 10.

Step 3:

y − 5 = 10(x − 2)
y − 5 = 10x − 20
y = 10x − 15