Question

In the following problem, begin by drawing a diagram that shows the relations among the variables. If w=3x

2
+2y
2
−z
2
and z=2x
2
+3y
2
, find a. (
∂y
∂w
​
)
z
​
. b. (
∂z
∂w
​
)
x
​
. c. (
∂z
∂w
​
)
y
​
. a. (
∂y
∂w
​
)
z
​
= b. (
∂z
∂w
​
)
x
​
= c. (
∂z
∂w
​
)
y
​
=

Answer 1

We can start by drawing a diagram that shows the relations between the variables:

x

|

|

2x²+3y²=z w = 3x²+2y²-z

|

|

z

a. To find ( ∂y/∂w )z, we need to differentiate z with respect to y, and then differentiate w with respect to y, and divide the two derivatives:

∂z/∂y = 6y

∂w/∂y = 4y

Therefore, ( ∂y/∂w )z = (∂z/∂y)/(∂w/∂y) = (6y)/(4y) = 3/2

b. To find ( ∂z/∂w )x, we need to differentiate z with respect to w, and then differentiate x with respect to w, and divide the two derivatives:

∂z/∂w = -6x

∂x/∂w = 6x

Therefore, ( ∂z/∂w )x = (∂z/∂w)/(∂x/∂w) = (-6x)/(6x) = -1

c. To find ( ∂z/∂w )y, we need to differentiate z with respect to w, and then differentiate y with respect to w, and divide the two derivatives:

∂z/∂w = -6x

∂y/∂w = 4y

Therefore, ( ∂z/∂w )y = (∂z/∂w)/(∂y/∂w) = (-6x)/(4y) = (-3x)/(2y)

Hence, ( ∂y/∂w )z = 3/2, ( ∂z/∂w )x = -1, and ( ∂z/∂w )y = (-3x)/(2y).