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).

Image 1: y=f(x)

At the point

the value of the function is:

And at the point

The value of the function is:

The following rate is called the average changing rate of the function.

The limit of this rate when

is called the derivative or the differential at this point and is equal to the slope of the tangent at this point:

The process of finding the derivative is called differentiation.

Notation

There are mainly three ways to note derivatives. It’s Leibniz’s, Lagrange’s and Newton’s way.

Leibniz’s notation:

The first derivative is:

The second derivative is:

For higher derivatives:

Lagrange’s notation:

The first derivative is:

The second derivative is:

For higher order derivatives:

Newton’s notation:

When the independent variable is time, Newton’s notation can be used. It cannot be used if the independent variable is not the time. For y=f(t),

The first derivative is:

The second derivative is:

Examples

Example 1

Let’s calculate the derivative of:

So,

The derivative is:

At x=3 the value of the derivative is:

Example 2

Let’s calculate the derivative of:

So,

The derivative is:

Derivatives in practice are calculated using rules and tables of results and not by the definition.

Continuity and Differentiation

If a function is differentiable at a point a, it will be continuous. But a function which is continuous at a point a, it is not necessary differentiable at this point.