## Summary

- A tangent is a line that just touches the curve but doesn’t go through it.

The gradient of a tangent = Gradient of a curve at that point - A normal is a straight line perpendicular (at right angle 90°) to a curve.

We know that differentiation is the process that we use to find the gradient of a point on the curve. However, we can also find the gradient of a curve at a given point by drawing a tangent at that point (which is a straight line) and finding the gradient of that line which is equal to the gradient/derivative of the curve as well.

#### What is a Tangent?

As shown in Fig 1, a tangent just touches the curve but doesn’t go through it. The line intersects the curve at a point and the gradient of the line is equal to the derivative of the curve at the point.

#### Example #1

Q. Find the equation of tangent to the curve *y = (6 – x)(x + 3)* at the point *(3, 5)*

*Solution:*

Write the equation in the form that we can differentiate:

Now differentiate it:

The gradient of the tangent is equal to gradient of the curve at point (3, 5):

At x=3

Hence *m(gradient) = -3* as well.

Now to find the equation of the tangent:

Point is (3, 5) and m = -3

**equation of tangent at point (3, 5)**

#### What is a Normal?

As shown in Fig 2, A normal is a straight line perpendicular (at right angle 90°) to a curve or to the tangent of the curve at that point.

If two lines, with gradients *m1* and *m2* are at right angles then .

Therefore if the gradient of the tangent is *m1* than the gradient of the normal is

#### Example #2

Q. Find the equation of the normal to the curve at the point where *x = 2*.

*Solution:*

Firstly find the y coordinate at *x=2:*

Now differentiate the equation of the curve to find the gradient of the tangent at:

At x=2

Hence *m1 = 8*

Find *m2:*

Equation of normal is where Point is (2, 2) and .

**equation of normal at point (2, 2)**

##### Reference

- CGP AS-Level Edexcel Mathematics complete revision and practice book