12.6PureStretch

Gradients, Tangents and Normals

The derivative gives the gradient of the tangent at any point. From that one number you can write the equation of the tangent — and, using perpendicular gradients, the equation of the normal.

30 min Video by Zeeshan Zamurred Differentiation

Edexcel AS Level Maths: 12.6 Gradients, Tangents and NormalsWatch the full walkthrough before the notes below.

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What you'll be able to do

Find the gradient of a tangent at a point
Write the equation of a tangent
Find the gradient of the normal
Write the equation of a normal

Tangent gradient and equation

Differentiate to get the gradient function, substitute the point’s $x$ -value for the tangent gradient $m$ , then use $y - y_{1} = m (x - x_{1})$ with the point on the curve.

y - y_{1} = m (x - x_{1}), m = \frac{d y}{d x}_{x = x_{1}}

The tangent through

(x_{1}, y_{1})

\frac{d y}{d x} = 2 x

; at

x = 2

m = 4

y - 4 = 4 (x - 2)

Answer

y = 4 x - 4

The normal

The $normal$ is perpendicular to the tangent at the point of contact. Its gradient is the $negative reciprocal$ of the tangent gradient.

m_{normal} = - \frac{1}{m _{tangent}}

Perpendicular gradients multiply to −1.

1Tangent gradient is

4

, so normal gradient is

- \frac{1}{4}

y - 4 = - \frac{1}{4} (x - 2)

Answer

y = - \frac{1}{4} x + 4.5

Tip — Find the tangent gradient first; the normal gradient is just its negative reciprocal.

Formula recap

m = \frac{d y}{d x}_{x = x_{1}}

Tangent gradient.

y - y_{1} = m (x - x_{1})

Equation of tangent / normal.

m_{normal} = - \frac{1}{m _{tangent}}

Normal gradient.

Common mistakes to avoid

Using the normal gradient as the tangent gradient.

The normal is the negative reciprocal of the tangent gradient.

Forgetting to find the y-coordinate of the point.

You need (x₁, y₁) on the curve to write the line equation.

Key takeaways

Tangent gradient = dy/dx evaluated at the point.
Use y − y₁ = m(x − x₁) for the tangent line.
Normal gradient = −1/(tangent gradient).

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