Unit – 4

Application of Partial Differentiation and Successive Differentiation

In this unit we are going to calculate the value of a maximum value and minimum value of a function with or without given relation between the variables. We  will also  calculate the nth order derivative of the given function. With the help of Leibnitz’s method we will differentiate and integrate a function simultaneously.

4.1 Maxima and Minima of  a function of two independent variables

As we know that the value of a function at maximum point is called maximum value of a function. Similarly the value of a function at minimum point is called minimum value of a function.

The maxima and minima of a function is an extreme biggest and extreme smallest point of a function in a given range (interval) or entire region. Pierre de Fermat was the first mathematician to discover general method for calculating maxima and minima of a function. The maxima and minima are complement of each other.

Maxima and Minima of a function of one variables

If  f(x) is a single valued function defined in a region R then

Maxima is a maximum point if and only if

Minima is a minimum point if and only if   

Maxima and Minima of a function of two independent variables

Let be a defined function of two independent variables.

Then the point is said to be a maximum point of if

Or   =

For all positive and negative values of h and k.

Similarly the point is said to be a minimum point of if

Or   =

For all positive and negative values of h and k.

Saddle point:Critical points of a function of two variables are those points at which both partial derivatives of the function are zero. A critical point of a function of a single variable is either a local maximum, a local minimum, or neither. With functions of two variables there is a fourth possibility - a saddle point.

A point is a saddle point of a function of two variables if

At the point.

Stationary Value

The value is said to be a stationary value of if

i.e. the function is a stationary at (a , b).

Rule to find the maximum and minimum values of

1. Calculate.
2. Form and solve , we get the value of x and y let it be pairs of values
3. Calculate the following values :

4.     (a) If

(b) If

(c) If

(d) If

Example1 Find out the maxima and minima of the function

Given     …(i)

Partially differentiating (i) with respect to x we get

    ….(ii)

Partially differentiating (i) with respect to y we get

    ….(iii)

Now, form the equations

Using (ii) and (iii) we get

          using above two equations

Squaring both side we get

Or 

This show that

Also we get

Thus we get the pair of value as

Now, we calculate

Putting above values in

At point (0,0)  we get

So, the point (0,0) is a saddle point.

At point we get

So the point is the minimum point where

In case 

So the point is the maximum point where

Example2 Find the maximum and minimum point of the function

Partially differentiating given equation with respect to and x and y then equate them to zero                 

On solving above we get 

Also

Thus we get the pair of values (0,0), (,0) and (0,

Now, we calculate

At the point (0,0)

So function has saddle point at (0,0).

At the point (

So the function has maxima at this point (.

At the point (0,

So the function has minima at this point (0,.

At the point (

So the function has an saddle point at (

Example3 Find the maximum and minimum value of

Let

Partially differentiating given function with respect to x and y and equate it to zero

  ..(i)

    ..(ii)

On solving (i) and (ii) we get

Thus pair of values are

Now, we calculate

At the point (0,0)

So further investigation is required

On the x  axis y  = 0  , f(x,0)=0

On the line y=x,

At the point

So that the given function has maximum value at

Therefore maximum value of given function

At the point

So that the given function has minimum value at

Therefore minimum value of the given function

4.2Lagrange’s method of Undetermined Multipliers

Generally  to calculate the stationary value of a function with some relation by converting the given function into the least possible independent variables and then solve them. When this method fail we use Lagrange’s method.

This method is used to calculate the stationary value of a function of several variables which are all not independent but are connected by some relation.

Let  be the function in the variable x, y and z which is connected by the relation

Rule: a) Form the equation

Where is a parameter.

b) Form the equation using partial differentiation is

(We always try to eliminate)

c) Solve the all above equation with the given relation

These give the value of

These value obtained when substituted in the given function will give the stationary value of the function.

Example1 Divide 24 into three parts such that the continued product of the first, square of second and cube of third may be maximum.                                    

Let first number be x, second be y and third be z.

According to the question

Let the  given function  be  f

And the relation

By Lagrange’s Method

   ….(i)

Partially differentiating (i) with respect to x,y and z and  equate  them  to zero

   ….(ii)

   ….(iii)

   ….(iv)

From (ii),(iii)  and (iv) we get

On solving

Putting it in given relation we get

Or 

Or

Thus the first number is 4 second is 8 and third is 12

Example2 The temperature T at any point   in space is  .Find  the  highest temperature on the surface of the unit sphere.

Given function is  

On the surface of unit sphere given    [ is an equation of unit sphere in 3   dimensional space]

By Lagrange’s Method

     ….(i)

Partially differentiating (i) with respect to  x, y and z and equate them to zero

or           …(ii)

or           …(iii)

          …(iv)

Dividing (ii) and (ii) by (iv) we get

Using given relation 

Or       

Or                

So that

Or 

Thus points are

The maximum temperature is

Example3  If ,Find the value of x and y for  which is maximum.

Given function is

And relation is

By Lagrange’s Method

[]     ..(i)

Partially differentiating (i) with respect to  x, y and z and equate them to zero

Or       …(ii)

Or       …(iii)

Or       …(iv)

On solving (ii),(iii) and (iv) we  get

Using the given relation  we get

So that

Thus the point for the maximum value of the given function is

Example4  Find  the points on the surface nearest to the origin.

Let be any point on the surface, then its distance from the origin is

Thus the given equation will be

And relation is

By Lagrange’s Method

     ….(i)

Partially differentiating (i) with respect to  x, y and z and equate them to zero

Or        …(ii)

Or        …(iii)

Or

Or

On solving equation  (ii) by (iii) we get

And     

On subtracting we  get

Putting in above

Or 

Thus

Using the given relation we  get

= 0.0 +1=1

Or 

Thus point on the surface nearest to the origin is

4.3.1 Successive Differentiation with  nth  derivative:

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:

Function	Derivaties
	…………..
	………..
	…………
	………….

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.

4.3.2Leibnitz’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

4.4 Leibnitz’s rule:

This method is used to differentiate the given function under integral sign.

Let be a function defined in two variables x and (is a parameter) .

Define

To  find the derivative of , when exist is not always possible to first evaluate this integral and then find the  derivative, Such type of problems are solved by using Leibnitz’s Rules.

Leibnitz’s First Rule

 This rule is applicable when the limits of the given integral are constant.

This rule helps us to calculate the value of the definite integrals which are difficult or even impossible to find out by other methods.

Note: The rule for differentiation under integral sign of an infinite integral is the same as for a definite integral.

Important To solve the integral problem many times we are going to use partial fraction method to split the fraction. Some standard form are given below:

S. No.	Form of rational fraction	Partial fraction form
1
2
3
4
5

Example1 Find the value of

Let     

Differentiating both side with respect to a we get

Using Leibnitz’s Rule

Now,  integrating both side

As given integral

Thus  

Example2 Differentiating under the integral sign ,find the value of

Let

Differentiating both side with respect to parameter 

Using Leibnitz’s theorem

On integrating both side

When

Hence

Leibnitz’s Second rule for variable limits of integration

This method is applicable when limits of integration are variable.

Example1 If ,prove that y satisfied the differential equation

Given

Differentiating both side with respect to x

By Leibnitz’s Rule

Again differentiating with respect to x we get

Using Leibnitz’s Rule

Hence proved

Example2 Verify by direct integration and prove that

Let

Differentiating both  side with respect  to a, we  get

By Leibnitz’s  theorem

+

Hence proved

Direct integration

By   product  of two  function  under integration

So that,