====== 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 <latex>y=x^3</latex>



<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 <latex>x</latex>?



===== Exercises =====



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



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

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

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