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.
