UNIT 2

Partial differentiation

Question-1: Find all the second order partial derivative;

z = x cos y – y cos x

Solution:       here, first we will find first order derivative,

=  cos y + y sin x      ,             = - x sin y – cos x

Now,

  = =  ) = y cos x

  = =     = - x cos y

= = (- x sin y – cos x) = - sin y + sin x

= = (cos y + y sin x ) = - sin y + sin x

Here we notice,       = 

Question-2: Find all of the second partial derivatives of the function:

f( x, y, z) = 3x³ sin z

Solution:   We notice here that, the function has three variables,

Now first we will find all of the first order partial derivatives ,

 = 9x² sin z   ,    =  - 3x³ sin z     ,    = 3x³ cos z

  = =  ) = 18 x² sin z  

  = =     =  3x³ sin z

= = () =

= = (9x² sin z ) =    - 9x² sin z  

= =  3x³ cos z) = - 3x³ sin z

= = (- 3x³ sin z ) = - 3x³ cos z

= = (9x² sin z) = 9x² cos z

= = (3x³ cos z) =  - 3x³ cos z

= = (3x³ cos z) =  9x³ cos z)

Question-3:    Verify Euler’s theorem for the following function:

u = x³ + y³ + 3 xy²

Solution:  Here u is homogeneous function of degree 3.

To verify,

x + y = nu

Here,

u = x³ + y³ + 3xy²

Differentiating partially with respect to x, we get

= 3x² + 0 + 3y² =  3x² + 3y²

Or   x   =  3x³ + 3xy²   ----------------------- (1)

Now , differentiating partially with respect to y, we get

= 0 + 3y² + 6xy

Or         y =   3y³ + 6xy² ---------------------------(2)

Adding equation (1) and (2), we get

x + y =  3x³ + 3xy² + 3y³ + 6xy² 

= 3x³ + 3y³ + 9xy²

= 3( x³ + y³ + 3xy² )

x + y   =  3u

Hence Euler’s theorem is verified.

Question-4:  If u = log ( x³ + y³ + z³ - 3xyz) , then show that

Solution:     here ,       log ( x³ + y³ + z³ - 3xyz)

Differentiate it partially with respect to x, we get

Similarly,

,  

And

Now adding all these results,

=   

=          

Now we will perform following steps in order to get the result,

=         

Question-5:  If  u =  , show that,

x + y 0

Solution :  here, differentiate u partially with respect to x,

Or,

Now partially differentiate w.r.t y,

y      =    +  

By adding,  we get

x + y 0           hence proved.

Question-6:  if ,

Prove that -      x + y 1

Solution:      we are given,

Here f is a homogeneous function of degree 1,

By Euler’s theorem,

Or     x + y 1.       Hence proved.

Question-7:  If,

Then prove that ,         x + y tan u

Solution:       it is given that,

Here f is a homogeneous function of degree ½

By using Euler’s theorem,

So,     x + y tan u.                        hence proved.

Question-8:  if  u =      , show that

,     x + y + z + 3 tan u = 0.

Solution:   it is given that,   u =  

Sin u = = f

Here f is a homogeneous function ,

By Euler’s theorem,  

x + y + z

So that,

x + y + z + 3 tan u = 0                                 hence proved.

Question– 9:   if ∅( x , y, z) = 0, then show that,

Solution:   here first we treat x as constant, we get

Now we treat y as constant,

Similarly,

By multiplying equation (1) , (2), and (3), we get

Hence proved.

Question-10:    if u = u(y –z, z – x, x – y)

Prove that,       x + y + z         

Solution :  here ,  u = u(y –z, z – x, x – y)

Suppose,  X = y-z ,  Y = z-x,     Z = x-y   ………………………. (i)

So,     u = u(X , Y , Z)

Now,

With the help of (i), (ii), (iii) (iv) gives,

Adding (v) , (vi) and (vii), we get,           

x + y + z                       

Hence proved.