Unit – 3
Multivariable calculus: Differentiation
Q1) Evaluate
A1)
Q2) Evaluate
A2)
1. 
2. 
Here f1 = f2
3. Now put y = mx, we get
Here f1 = f2 = f3
Now put y = mx²
4. 
Therefore,
F1 = f2 = f3 =f4
We can say that the limit exists with 0.
Q3) Calculate 
and 
for the following function
f (x, y) = 3x³-5y²+2xy-8x+4y-20
A3) To calculate 
treat the variable y as a constant, then differentiate f (x, y) with respect to x by using differentiation rules,
= 
[3x³-5y²+2xy-8x+4y-20]
= 
3x³] - 
5y²] + 
[2xy] -
8x] +
4y] - 
20]
= 9x² - 0 + 2y – 8 + 0 – 0
= 9x² + 2y – 8
Similarly partial derivative of f (x, y) with respect to y is:
= 
[3x³-5y²+2xy-8x+4y-20]
= 
3x³] - 
5y²] + 
[2xy] -
8x] +
4y] - 
20]
= 0 – 10y + 2x – 0 + 4 – 0
= 2x – 10y +4.
Q4) If 
, then show that- 
A4) Here we have,
u = 
…………………. (1)
now partially differentiate eq. (1) w.r.t to x and y, we get
= 
Or
………………. (2)
And now,
Adding eq. (1) and (3), we get
Hence proved.
Q5) Find the directional derivative of 1/r in the direction 
where 
A5) Here 
Now,
And 
We know that-
So that-
Now,
Directional derivative-
Q6) Find the directional derivative of 1/r in the direction 
where 
A6) Here 
Now,
And 
We know that-
So that-
Now,
Q7) Find the directional derivative of
At the points (3, 1, 2) in the direction of the vector 
.
A7) Here it is given that-
Now at the point (3, 1, 2)-
Let 
be the unit vector in the given direction, then
at (3, 1, 2)
Now,
Q8) Find 
if u = x³y⁴ where x = t³ and y = t².
A8) As we know that by definition, 
= 
3x²y⁴3t² + 4x³y³2t = 17t¹⁶.
Q9) If z is the function of x and y , and x = 
, y = 
, then prove that,
A9) Here , it is given that, z is the function of x and y & x , y are the functions of u and v.
So that,
……………….(1)
And,
………………..(2)
Also there is,
x = 
and y = 
,
Now,
, 
, 
, 
From equation(1) , we get
……………….(3)
And from eq. (2) , we get
…………..(4)
Subtracting eq. (4) from (3), we get
= 
)
– (
= x
Hence proved.
Q10) If 
then prove that grad u, grad v and grad w are coplanar.
A10)
Here- 
Now-
Apply 
Which becomes zero.
So that we can say that grad u, grad v and grad w are coplanar vectors.
Q12) If 
then find the divergence and curl of 
.
A12)
We know that-
Now-
