# 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.

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)****.**

### Derivative of some standard functions

### Algebra of Derivative of functions

If ** f(x)** and

**be any two functions such that their derivatives are defined over the common domain then**

*g*(*x*)- [f(x)+g(x)]=f(x)+g(x)
- [f(x)-g(x)]=f(x) – g(x)
- [f(x)g(x)]=g(x)f(x) + f(x)g(x)

This is also known as **Product Rule of Differentiation.**

- = where g(x) 0.

This is also known as **Quotient Rule of Differentiation**

### **Examples**

**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 2: **Find 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*.

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

#### Answer Key

- 15 + 20
- -Sinx – 3 x
- Cosx(Tan x)+Sin x( x)
- 5 + 34

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