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Trace: » differentiation


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 y=x^3

<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 x?

Exercises

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

1) y=x^2+1

ANSWER: \frac{dy}{dx}=2x

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