UNIT 1

DIFFERENTIAL CALCULUS

1.1 Successive differentiation

It  is  the  process  of differentiating the given  function simultaneously many times and  the result   obtained are  called successive  derivative.

Let be a differentiable function.

First derivative is denoted by

Second derivative is  

Third derivative is

Similarly the nth derivative is

Example:

Example1:  Find  the nth  derivative of  

Since  

Differentiating  both side with  respect to  x

[

Again  differentiating  with respect to x

Again  differentiating  with respect to x

   Similarly the  nth  derivative is

Example2:  Find the  nth  derivative of

    Let 

]

Differentiating  with  respect  to   x  we get

Again  differentiating  with  respect to x  we get

Again  differentiating  with  respect to x  we get

Similarly Again  differentiating  with  respect to x  we get

Example3: Find the nth  derivative

  Let

   Differentiating with respect to x.

  Again differentiating with respect to x.

    Again differentiating with respect to x.

    Again differentiating with respect to x.

    Again differentiating with respect to x.

  Similarly the  nth derivative with respect to  x.

1.2 Leibnitz‟s theorem

If u and v are the function of x  such  that  their nth  derivative exists, then the nth derivative of  their product will  be

Example1:Find  the nth derivative of  

  Let

Also

   By Leibnitz’s theorem

…(i)

Here 

Differentiating with  respect  to  x, we get

Again  differentiating  with respect to x, we get

Similarly  the  nth derivative will be

From  (i) and (ii) we have,

Example2:  If ,     then show that

   Also, find 

Here

Differentiating with respect to x, we get

…(ii)

Squaring   both  side  we get

…(iii)

Again differentiating with  respect to  x ,we get

Using Leibnitz’s  theorem  we  get

…(iv)

      Putting x=0  in equation (i),(ii) and (iii)  we get

Putting  n=1,2,3,4….

      ………………

   Hence   

Example3: If then show that 

   Given 

          Differentiating both side  with  respect to  x.

…..(ii)

Again differentiating with  respect to  x, we get

…(iii)          

By  Leibnitz’s theorem

…(iv)

Putting x=0 in equation  (i),(ii),(iii) and (iv) we  get

 Putting n=1,2,3,4… so  we get

Hence we  have

1.3 Taylor‟s and Maclaurin‟s theorem

Taylor’s theorem:

If (i) f(x) and its first (n-1) derivative be continuous in [a, a+h],

(ii) exist for every value of x in (a, a+h), then there is  at least one number such that

This is called Taylor’s theorem with Lagrange’s form of remainder

Taylor’s Series:

If can be expanded as an infinite series, then

If possesses derivative of all orders and  the  remainder .

Corollary: Taking and in equation (i) we get

Taking in above we get Maclaurin’s series.

Example1: Expand the polynomial in power of , by Taylor’s theorem.

Let .

Also

Then

Differentiating with respect  to x.

Again differentiating with respect to x the above function.

Again differentiating with respect to x the above function.

Also the value of above functions at x=2 will be

By Taylor’s theorem

On substituting above values we get

Example2: Expand in power of

Let

 Also  

Differentiating f(x) with respect to x.

Again differentiating f(x) with respect to x.

Again differentiating f(x) with respect to x.

Also the value of above functions at x=1 will be

By Taylor’s theorem

On substituting above values we get

=

Example3:Expand in power of. Hence find the value of correct to  four decimal places.

Let

and .

Differentiating with respect to x.

Again differentiating with respect to x.

Again differentiating with respect to x.

Again differentiating with respect to x.

Also the value of above functions at will be

By Taylor’s theorem

On substituting above values we get

At

.

Maclaurin’s theorem:

This is a particular case of Taylor’s theorem in which a=0 and h=x in  Taylor’s theorem.

If f(x) can be expanded as an infinite series, then

Where the remainder is 

Example1:Ifusing Taylor’s theorem, show that for .

Deduce that

Let then

Differentiating with respect to x.

  .Then

Again differentiating with respect to x.

  Then  

Again differentiating with respect to x.

  Then  

By Maclaurin’s theorem

Substituting the above values we get

Since

Hence   

Example2: Prove that

Let 

Differentiating   with respect to x.

Again differentiating with respect to x.

Again  differentiating with respect to x.

Again differentiating with  respect to x.

and so on.

Putting , in above derivatives we get

so on.

By Maclaurin’s theorem

+………

Substituting the above values we get

Example3:Prove  that

 Let

Differentiating above function with respect to x.

Again differentiating above function with  respect to x.

Again differentiating above function with  respect to x.

Again differentiating above function with  respect to x.

Putting , in above derivatives we get

so on.

By Maclaurin’s theorem

+………

Substituting the above values we get

Example: Expansion of Some standard series

Let and

Differentiating above function with respect to x.

By Maclaurin’s theorem

+………

Substituting the above values we get

2.    

Let and

Differentiating above function with respect to x.

By Maclaurin’s theorem

+………

Substituting the above values we get

3.

Let and

Differentiating above function with respect to x.

.

 By Maclaurin’s theorem

+………

Substituting the above values we get

4.

Let and also

Differentiating above function  with respect to x.

By Maclaurin’s theorem

+………

Substituting the above values we get

5. 

Let and also

Differentiating above function  with respect to x.

By Maclaurin’s theorem

+………

Substituting the above values we get

6.

Let and also

Differentiating above function with respect to x.

By Maclaurin’s theorem

+………

Substituting the above values we get

7.

Let

Differentiating above function  with respect to  x.  

By Maclaurin’s theorem

+………

Substituting the above values we get

++………

+………

8. 

Let 

Differentiating above function with respect to  x.

By Maclaurin’s theorem

+………

Substituting the above values we get

+………

+………

9. 

  Let 

Differentiating above function with respect to  x.

By Maclaurin’s theorem

+………

Substituting the above values we get

+………

+………)

10. 

Let 

Differentiating the above function with respect to x.

  By Maclaurin’s theorem

+………

Substituting the above values we get

1.4 Indeterminate forms

Let   are functions of x  only and have  zero limit when  .(where a is a certain point)

is called the  indeterminate form. It   means that   does not exist.

The indeterminate form is distributed in the following form:

1. Form  0/0:

If   then

This is called L-Hospital Rule.

In case 

Then   

Differentiation of numerator and denominator are done separately as many times as required.

Example1: Evaluate the limits of

As we can see that

Therefore 

Using L-Hospital rule

Again using  L-Hospital Rule

Again using  L-Hospital Rule

Hence

Example2:Evaluate the limits of

Since 

  Therefore 

Using L-Hospital rule

Again using  L-Hospital Rule

Hence

Example3:Evaluate

Since form     ….(i)

Consider , let  

Taking log on both side

Differentiating both side with respect to x we get

Or

Or   ….(ii)

Using L-Hospital Rule   in (i) we have

[using (ii)]

=0/0   form

Again Using L-Hospital Rule                           

form

where

Again Using L-Hospital Rule

=

= form

Again using L-Hospital rule

=

Hence   .

2. Form :

If   then

This is called L-Hospital Rule.

In case 

.

Then   

Differentiation of numerator and denominator are done separately as many times as required. We use series and standard limits.

Example1: Find 

Since form

Using L-Hospital Rule 

= 

=  form

Again Using L-Hospital Rule

=

Hence

Example2: Find 

Since form

Using L-Hospital Rule  

= form

Again using L-Hospital Rule

= form

Again Using L-Hospital Rule

=

Hence .

Example3: Evaluate the limit

Since form

Therefore form

Using L-Hospital Rule 

form

Again Using L-Hospital Rule

Hence

Forms  reducible to 0/0  or

form:

Form

:   If  

then

Then will be of the form 0/0 when  .

 And will be of the form when 

b.     Form : If   then

Then

c.      Forms :   If  form.

Let

Taking log on both side  we get

This is solved by the above method then we have will give.

Example1:   Evaluate the limit

  Since  form

         Let  

            Taking log on both side we get

form

Using L-Hospital Rule

form

Again using L-Hospital Rule

=

=

Here

Hence .

Example2:  Evaluate the limit

Since  form

Let 

  Taking log on both side we get

form

Using L-Hospital Rule we get

Here 

Hence.

Example3:Evaluate the limit

Since 

Let y = 

Taking log on both side we get

form

Using L-Hospital Rule we get

  So,

Hence

Reference Books:

1. A text book of Applied Mathematics Volume I and II by J.N. Wartikar and P.N. Wartikar

2. Higher Engineering Mathematics by Dr. B. S. Grewal

3. Advanced Engineering Mathematics by H. K. Dass

4. Advanced Engineering Mathematics by Erwins Kreyszig