# Definition of Differentiation

### Introduction

Its morning. You get into your car and you drive to your work. Its 50km away and you need 1 hour to get there. So your average speed is u=s/t=50Km/hour. But when you drive in the city you drive slowly and when you drive in the highway you drive quickly. In order to calculate your speed at any point, the “instant” speed, you need to divide a small part of the trip that you are interested in calculating by the time you need to travel over it. So if you travel 1Km in 0.01 hours (36 seconds) your local speed will be u=1/0.01=100Km/hour. The smaller the interval you are calculating, the more accurate your calculation will be.

In a more formal way, consider the function y=f(x).