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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.

differention

https://goo.gl/images/vrhn3e

Volume V of the spherical balloon is given by, V=\frac{4}{3}\Pi r^{2}.

If r=1 cm then V = \frac{4}{3}\Pi cm^{3}.

If r=2 cm then V= \frac{32}{3}\Pi cm^{3}.

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 \frac{dV}{dr} or V^{{1}}(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 f^{{1}}(c) and it is defined by f(c)=lim\left ( \frac{f(c+h)-f(c)}{h} \right ) provided this limit exists.

 

II. Derivative of a function in general.

In general, the derivative of f(x) is denoted by f^{{1}}(x) and it is defined by

f^{{'}}(x)=lim\left ( \frac{f(c+h)-f(c)}{h} \right ) 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 \frac{dy}{dx} or y^{{'}} or f^{{'}}(x).

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

To differentiate a function f^{{'}}(x) means to find its derivativef^{{'}}(x).

 

Derivative of some standard functions

differention1

 

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. \frac{d}{dx}[f(x)+g(x)]=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)
  2. \frac{d}{dx}[f(x)-g(x)]=\frac{d}{dx}f(x) – \frac{d}{dx}g(x)
  3. \frac{d}{dx}[f(x)g(x)]=g(x)\frac{d}{dx}f(x) + f(x)\frac{d}{dx}g(x)

This is also known as Product Rule of Differentiation.

  1. \frac{d}{dx} \left [ \frac{f(x)}{g(x)} \right ]=\frac{g(x)\frac{d}{dx}f(x)-f(x)\frac{d}{dx}g(x)}{\left [ g(x) \right ]^2} where g(x) \neq 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)=x^{{11}} + 3x.

f^{{1}}(x)=\frac{d}{dx}(x^{{11}}) +3\frac{d}{dx}(x)=11x^{{10}}+3(1)=11x^{{10}}+3

 

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

f^{{1}}(x)=7\frac{d}{dx}(Sin x) -3\frac{d}{dx}(Tan x)=7Cos x – 3Sec^{{2}} x

 

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

f^{{1}}(x)=\frac{d}{dx}(Sin xCos x)

Cos x\frac{d}{dx}(Sin x) + Sin x\frac{d}{dx}(Cos x)= Cos x(Cos x) + Sin x(-Sin x)using Product rule.

=Cos^{{2}} x – Sin^{{2}} x

Example 4: Find the derivative of f(x)= \frac{Sin x}{Cos x}.

f^{{'}}(x)= \frac{d}{dx}\left ( \frac{Sin x}{Cos x} \right )

=\frac{Cosx\frac{d}{dx}(Sinx)-Sinx\frac{d}{dx}(Cosx)}{Cos^2x}\frac{Cos x(Cos x)-Sin x(-Sin x)}{Cos^2x} using Quotient Rule.

=\frac{Cos^2x+Sin^2x}{Cos^2x}=\frac{1}{Cos^2x}

 

Example 5: Find the derivative of f(x)=e^x+20x^12+15.

f^1(x)=\frac{d}{dx}(e^x)+20\frac{d}{dx}(x^12)+\frac{d}{dx}(15)=

=e^{{x}} +20(12x^{{11}})+0=e^x+240x^{{11}}.

 

differention2

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

 

  1. f(x)=x^{15}+20x
  2. f(x)=Cosx -3Tan x
  3. f(x)=Sinx Tanx
  4. f(x)=\frac{Cosx}{Sinx}
  5. f(x)=5e^{{x}} + 2x^{{17}} + 27

Answer Key

 

  1. 15x^{{4}} + 20
  2. -Sinx – 3Sec^{{2}} x
  3. Cosx(Tan x)+Sin x(Sec^{{2}} x)
  4. \frac{-1}{Sin^2 x}
  5. 5e^x + 34x^16