Differentiation from First Principles

In the last investigation you conjectured that the gradient of a tangent to the curve at any point of the graph was . To justify this conjecture we will 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?

Question 2

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

Question 3

Expand and simplify your expression. What happens now as

Question 4

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