Answer:
We can start by drawing a diagram that shows the relations between the variables:
x
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2x²+3y²=z w = 3x²+2y²-z
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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).