Differentiation from first principles

Graph 1: Decreasing the horizontal distance between two points

The horizontal distance between the two points is given by

. Use the slider to change the value of

. Decreasing the value of

will bring the two points closer together. You will notice as

decreases and approaches zero, the line between the two points approaches the tangent of the curve at

. Therefore we can say that as

, the gradient of the secant will approach the gradient of the function at

Use the slider to change the value of

, moving the point (

) on the graph. Moving the slider for

, we can see that the the above holds true for any point

In the graph below we have a curve

. The point A is fixed at

, and therefore has a y-coordinate of

. The point B is at the point where

, and therefore has a y-coordinate of

. The gradient of the secant is therefore given by:

Therefore to generalise for any function

: At any

, when

, as

Move the point B towards A to see how the line through the secant approaches the tangent and watch how the values change.