Tangent and Normal to Curve

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

If:

then the gradient of the tangent at � is:

B. Meaning of a Normal

The normal is the straight line perpendicular to the tangent at the point of contact.

If the gradient of the tangent is �, then:

Therefore:

Special Case

If the tangent is horizontal:

then the normal is a vertical line, so its gradient is undefined.

C. Equation of a Straight Line

Recall:

This is the point-gradient form of a straight-line equation.

To find the equation of a tangent:

Find the point on the curve.

Differentiate the curve.

Substitute the �-coordinate into �.

Obtain the tangent gradient.

Use �.

D. Derivation of the Tangent Equation

Suppose:

and the point of contact is:

The gradient of the curve at � is:

Using the equation of a straight line:

Therefore, the equation of the tangent is: