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# Differentiation

### Have you ever inflated a spherical balloon?

What do we observe? As we start inflating the balloon its radius starts increasing and consequently, its volume also starts increasing.

https://goo.gl/images/vrhn3e

Volume V of the spherical balloon is given by, V=.

If r=1 cm then V = .

If r=2 cm then V= .

If we want to find the rate at which the volume V increases as the radius r increases, then we want to find the rate of change of Volume with respect to the radius.

So, if we want to find the rate of change of a dependent variable (Volume V here) with respect to independent variable (radius r here), we want to find  or (r).

Here we will introduce the concept of Differentiation.

### I. Derivative of a function at a point.

Let f(x) be a real valued function and x = c be any point in its domain, then the derivative of f(x) at x= c is denoted by (c) and it is defined by f(c)=lim provided this limit exists.

### II. Derivative of a function in general.

In general, the derivative of f(x) is denoted by (x) and it is defined by

(x)=lim provided this limit exists.

Also, if y = f(x), then the derivative or the differential coefficient of f(x) with respect to x is denoted by  or  or (x).

The process of finding the derivative of a function is called Differentiation.

To differentiate a function (x) means to find its derivative(x).

### Algebra of Derivative of functions

If f(x) and g(x) be any two functions such that their derivatives are defined over the common domain then

1. [f(x)+g(x)]=f(x)+g(x)
2. [f(x)-g(x)]=f(x) – g(x)
3. [f(x)g(x)]=g(x)f(x) + f(x)g(x)

This is also known as Product Rule of Differentiation.

1.  = where g(x)  0.

This is also known as Quotient Rule of Differentiation

### Examples

Now let’s consider some examples on differentiation .

Example 1: Find the derivative of f(x)= + 3x.

(x)=() +3(x)=11+3(1)=11+3

Example 2Find the derivative of f(x)=7Sin x-3Tan x.

(x)=7(Sin x) -3(Tan x)=7Cos x – 3 x

Example 3: Find the derivative of f(x)=Sin xCos x .

(x)=(Sin xCos x)

Cos x(Sin x) + Sin x(Cos x)= Cos x(Cos x) + Sin x(-Sin x)using Product rule.

= x –  x

Example 4: Find the derivative of f(x)= .

(x)=

= using Quotient Rule.

==

Example 5: Find the derivative of f(x)=+20+15.

(x)=()+20()+(15)=

= +20(12)+0=+240.

## Check Point

Find the derivative of the following functions f(x) with respect to x.

1. f(x)=+20x
2. f(x)=Cosx -3Tan x
3. f(x)=Sinx Tanx
4. f(x)=
5. f(x)=5 + 2 + 27

1. 15 + 20
2. -Sinx – 3 x
3. Cosx(Tan x)+Sin x( x)
4. 5 + 34

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