# 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?