Solve the problem.
-$$\text { Evaluate } \frac { \partial \mathrm { z } } { \partial \mathrm { u } } \text { at } ( \mathrm { u } , \mathrm { v } ) = ( 4,5 ) \text { for the function } \mathrm { z } = \mathrm { xy } - \mathrm { y } ^ { 2 } ; \mathrm { x } = \mathrm { u } - \mathrm { v } , \mathrm { y } = \mathrm { uv }$$
A) 9
B) -1
C) 19
D) -185

Question

Quizplus · Accepted Answer

Using the chain rule, we have 
$$\frac{\partial z}{\partial u}=\frac{\partial z}{\partial x}\cdot\frac{\partial x}{\partial u}+\frac{\partial z}{\partial y}\cdot\frac{\partial y}{\partial u}$$
Plugging in $x=u-v$ and $y=uv$, we have
$$\frac{\partial z}{\partial x} = y = uv$$
$$\frac{\partial z}{\partial y} = x - 2y = (u-v) - 2uv$$
$$\frac{\partial x}{\partial u} = 1, \quad \frac{\partial y}{\partial u} = v$$
Plugging in the given values $(u,v)=(4,5)$, we have 
$$\frac{\partial z}{\partial u} = (uv)\cdot 1 + [(u-v)-2(uv)]\cdot (5) = -7uv-4= -7\cdot 4 \cdot 5 - 4 = \boxed{- 185}.$$

Solve the Problem A) 9 B) -1 C) 19 D) -185

Solve the Problem A) 9
B) -1
C) 19
D) -185