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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).

Derivative of some standard functions 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 . 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. 5 + 34  Our mission is to provide high quality online tutoring services, using state of the art Internet technology, to school students worldwide.

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Site by Little Red Bird Our mission is to provide high quality online tutoring services, using state of the art Internet technology, to school students worldwide.

Connect with us
+1-269-763-4602
+1-269-763-5024

Online test prep and practice
SCAT
CogAT
SSAT
ISEE
PSAT
SAT
ACT
AP Exam

Science Tutoring
Physics Tutoring
Chemistry Tutoring
Biology Tutoring

English Tutoring
Writing
Grammar