Use the Chain Rule to find $\frac { \partial w } { \partial r }$ and $\frac { \partial w } { \partial t }$ if $r = 5$ $s = 2$ and $t = 0$ $w = \frac { x ^ { 2 } y } { z ^ { 2 } } , \quad x = r e ^ { s t } , \quad y = s e ^ { n t } , \quad z = e ^ { r s t }$
A) $\frac { \partial w } { \partial r } = 20 , \quad \frac { \partial w } { \partial t } = - 550$
B) $\frac { \partial w } { \partial r } = 100 , \quad \frac { \partial w } { \partial t } = 200$
C) $\frac { \partial w } { \partial r } = 10 , \quad \frac { \partial w } { \partial t } = 0$
D) $\frac { \partial w } { \partial r } = 7 , \quad \frac { \partial w } { \partial t } = - 550$

Question

Quizplus · Accepted Answer

The partial derivatives are calculated using the chain rule. For $\frac{\partial w}{\partial r}$, differentiate $w$ with respect to $x$, $y$, and $z$ and then multiply by the derivatives of $x$, $y$, and $z$ with respect to $r$. Similarly, for $\frac{\partial w}{\partial t}$, differentiate $w$ with respect to $x$, $y$, and $z$ and multiply by the derivatives of $x$, $y$, and $z$ with respect to $t$. After substituting $r = 5$, $s = 2$, and $t = 0$, the calculations yield $\frac{\partial w}{\partial r} = 20$ and $\frac{\partial w}{\partial t} = -550$.

Use the Chain Rule to Find ∂w∂r\frac { \partial w } { \partial r }∂r∂w​

Use the Chain Rule to Find $\frac { \partial w } { \partial r }$