Provide an appropriate answer.
-Find $\frac { \partial \mathrm { w } } { \partial \mathrm { u } }$ when $\mathrm { u } = 2$ and $\mathrm { v } = - 5$ if $\mathrm { w } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = \frac { \mathrm { xy } } { \mathrm { z } } , \mathrm { x } = \frac { \mathrm { u } } { \mathrm { v } } , \mathrm { y } = \mathrm { u } + \mathrm { v }$, and $\mathrm { z } = \mathrm { u } \cdot \mathrm { v }$.
A) $\frac { \partial w } { \partial u } = - \frac { 3 } { 25 }$
B) $\frac { \partial \mathrm { w } } { \partial \mathrm { u } } = - \frac { 15 } { 4 }$
C) $\frac { \partial w } { \partial u } = \frac { 12 } { 25 }$
D) $\frac { \partial \mathrm { w } } { \partial \mathrm { u } } = - \frac { 6 } { 25 }$

Question

Quizplus · Accepted Answer

Using the chain rule, 
$$\frac{\partial w}{\partial u} = \frac{\partial w}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial w}{\partial y} \frac{\partial y}{\partial u} + \frac{\partial w}{\partial z}\frac{\partial z}{\partial u}$$
Firstly,  
$$\frac{\partial w}{\partial x} = \frac{y}{z}$$
$$\frac{\partial x}{\partial u} =\frac{1}{v}$$
Hence, 
$$\frac{\partial w}{\partial x} \frac{\partial x}{\partial u}=\frac{y}{vz}$$
Secondly, 
$$\frac{\partial w}{\partial y} =\frac{x}{z} $$ 
$$\frac{\partial y}{\partial u} = 1$$
Hence, 
$$\frac{\partial w}{\partial y} \frac{\partial y}{\partial u} = \frac{x}{z}$$
Thirdly,  
$$\frac{\partial w}{\partial z} =-\frac{xy}{z^2}$$
$$\frac{\partial z}{\partial u} =v$$
Hence, 
$$\frac{\partial w}{\partial z}\frac{\partial z}{\partial u}=-\frac{xy}{z}\cdot v=-uv=-2v^2$$
Therefore, 
\begin{align*}
\frac{\partial w}{\partial u} &=\frac{y}{vz}+\frac{x}{z}-2v^2\cdot\left(-\frac{5}{2}ight)\
&=\frac{u+v}{uv}\cdot\left(\frac{u}{v}ight)-25\
&=\frac{u^2+v^2}{uv^2}-25\
&=\frac{(2)^2+(-5)^2}{(2)\cdot(-5)^2}-25\
&=-\frac{6}{25}
\end{align*}

Provide an Appropriate Answer ∂w∂u\frac { \partial \mathrm { w } } { \partial \mathrm { u } }∂u∂w​

Provide an Appropriate Answer $\frac { \partial \mathrm { w } } { \partial \mathrm { u } }$