Differentiation from First Principles

From the Introductory Problem (TB p2), we can conclude that the gradient of a tangent to the curve at any point of the graph

was

. In this activity, we consider the gradient of a chord from the point

on the graph of

to a point that is a distance h further along the x-axis as shown on the graph below.

Question 1

a) What do you notice about the gradient of the chord as h decreases? b) What will happen as h approaches zero?

a) Explain why an expression for the gradient of the chord between the two points and b) What would happen if you let ?

Expand and simplify your expression. What happens now as

How does your answer to this compare to the conjecture you made for the gradient of in the first Geogebra investigation?