# Tangents

To construct the tangent to a curve at a certain point A, you draw a line that follows the general direction of the curve at that point. An example of this can be seen below.

Once the tangent is found you can use it to find the gradient of the graph by using the following formula:

where and are any two points on the tangent to the curve.

Question

Estimate the gradient to the curve in the graph below at point A.

First draw the tangent at the point given.

Select any two points on the tangent. The coordinates that we are using are (1, 0) and (2.5, 2000). Then use the formula below:

It is useful to remember that all lines and curves that slope upwards have a positive gradient.

All lines and curves that slope downwards have a negative gradient.

### Example

We want to find the gradient of the curve at .

First draw the tangent at .

Select any two points on the tangent. The coordinates that we are using are (-4, 9) and (0, -3).

where and are any two points on the tangent to the curve.