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# Introduction

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## Introduction

**Differentiation is a tool of mathematics that is primarily used for calculating rates of change. **

In Mechanics, the rate of change of **displacement** (with respect to time) is the **velocity** and the rate of change of **velocity** (with respect to time) is the **acceleration**.

When illustrating a function on a graph the rate of change is represented by the gradient.

**This means that the main aim of differentiation is to find the gradient at a specific point on a graph.**

Very often we know the equation for a graph (e.g. y = 3x^{2}). The graph of this equation looks like this:

As you can see, the gradients at the points **a**, **b** and **c** are quite different...

At GCSE level we estimate the gradient of a curve by drawing a tangent. **Differentiation** allows us to find the exact gradient at any of these points.

**Imagine two points on any curve where x is a function of y (written y = f(x)):
**

If we connect points **b** and **a** with a straight line, sometimes called a **chord** (as shown by the dotted line) then the gradient of this line is a very rough approximation to the gradient of the curve.

If the co-ordinates of **b** are (x_{1}, y_{1}) and the co-ordinates of **a** are (x_{2}, y_{2}) then the gradient of the line

The closer the points **a** and **b** are to each other, the more accurately we can measure the gradient of the line.

Using the graph above, but making the gap between **a** and **b** minimal, we can re-write our gradient:

You can express the position of **a** by saying it is the position of **b** plus an extra δx along the x-axis and an extra δy along the y-axis,

i.e. a = (x_{1} + δx , f (x_{1} + δx))

As **a** moves closer to **b**, δx becomes smaller and the chord ab becomes the tangent to the line at **a** (and therefore an exact measurement of the gradient of the line at **a**).

and that the correct value for the gradient occurs as δx → 0, we get the formula:

or,

This idea is differentiation from first principles and is useful to know, though the rules that result from this are the skills that are normally tested at A level.

**An example of differentiation from first principles**

Find the gradient of the function y = 3x^{2} when x = 5.

**Therefore:**

when x = 5,

So the gradient of the function y = 3x^{2} is 30 at (5, 75).