Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.41

Now differentiating u w.r.t.x (treating y as a constant) we find

fy u (x 4xy 3y ) (x ) (4xy) (y ) 2x 4y 0 2x 4y.^2222

x x x x x

∂ =∂ + + = ∂ +∂ +∂ = + + = +

∂ ∂ ∂ ∂ ∂

Again (treating x as a constant)

fy u (x 4xy 3y )^22 (x ) (4xy) (y )^22

y y y y y

∂ = ∂ + + = ∂+ + ∂ + ∂

∂ ∂ ∂ ∂ ∂

= 0 + 4x + 2y = 4x + 2y.

Example 89: If u = x^4 y^3 z^2 + 4x + 3y + 2z find ∂∂u u ux y z, , .∂∂ ∂∂

Solution:

u 4x y z 4 3 3 2

x

∂ = +

∂ ,

u 3y x z 3 2 4 2

y

∂ = +

∂ ,

u 2zx y 2. 4 3

z

∂ = +

∂

Example 90: If u = log (x^2 + y^2 ); find

2 2

2 2

u u,.

x y

∂ ∂

∂ ∂

2 2

u 1 .2x

x x y

∂ =

∂ + ,

2 2 2 2 2

2 2 2 2 2 2 2

u (x y ).2 2x(2x) 2(y x )

x (x y ) (x y )

∂ = + − = −

∂ + +

( ) ( )

( )

( )

( )

2 2 2 2 2

2 2 2 2 22 2 22

u 2y , u x y 2 2y 2y 2 x y

y x y y x y x y

∂ = ∂ = + − = −

∂ + ∂ + +

Example 91:
For f(x, y) = 3x^2 – 2x + 5, find fx and fy
Solution:
fx = 6x – 2 and fy = 0.
Example 92 :

If f(x, y) = 3 3
x y
y x− , find fx and fy

Solution:

x 3 4

f 1 3y

=y x+ y 4 3

f 3x 1

= −y x−

Example 93:
Find the first partial derivatives of f(x, y, z) = xy^2 z^3
Solution:
fx = 2 – u fy = z + 2uy fx = 3xy^2 z^2