====== Understanding Differentiation ======

===== Introduction =====

This article assumes that you are familiar with straight line graphs, the idea that a line has a gradient and that a gradient represents a rate of change. You also need to understand how use algebra and how to manipulate algebraic equations. If you are happy then read on...

===== Zooming In On A Curve =====

The diagram below shows the graph of the function <math>y=x^3</math>

<insert diagram here>

We are asked the question: how do we find the slope, otherwise known as the //rate of change//, of the graph at a particular value of <math>x</math>?

===== Exercises =====

Find the derivative (the gradient function) of the following (the first one has been done for you)...

1) <math>y=x^2+1</math>

**ANSWER:** <math>\frac{dy}{dx}=2x</math>

2) <math>y=2x^2+3x+1</math>\\
3) <math>y=x^5+12</math>\\
4) <math>y=2x^4-3x^3+x^2-3x+1</math>\\
5) <math>y=10x^4+3x^2+12</math>\\
6) <math>y=x^{-4}+1</math>\\
7) <math>y=-5x^{-2}+3x^2</math>\\
8) <math>y=x^{-3}-3x^{-2}</math>\\
9) <math>y=-x^{-1}+2</math>\\
10) <math>y=-5x^2+5x^2+100</math>