Write a chain rule formula for the following derivative.
-$$\frac { \partial \mathrm { w } } { \partial \mathrm { t } } \text { for } \mathrm { w } = \mathrm { f } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) ; \mathrm { x } = \mathrm { g } ( \mathrm { r } , \mathrm { s } , \mathrm { t } ) , \mathrm { y } = \mathrm { h } ( \mathrm { r } , \mathrm { s } , \mathrm { t } ) , \mathrm { z } = \mathrm { k } ( \mathrm { r } , \mathrm { s } , \mathrm { t } )$$
A) $\frac { \partial \mathrm { w } } { \partial \mathrm { t } } = \frac { \partial \mathrm { w } } { \partial \mathrm { x } } \frac { \partial \mathrm { x } } { \partial \mathrm { t } } + \frac { \partial \mathrm { w } } { \partial \mathrm { y } } \frac { \partial \mathrm { y } } { \partial \mathrm { t } }$
B) $\frac { \partial \mathrm { w } } { \partial \mathrm { t } } = \frac { \partial \mathrm { w } } { \partial \mathrm { x } } \frac { \partial \mathrm { x } } { \partial \mathrm { r } } + \frac { \partial \mathrm { w } } { \partial \mathrm { y } } \frac { \partial \mathrm { y } } { \partial \mathrm { s } } + \frac { \partial \mathrm { w } } { \partial \mathrm { z } } \frac { \partial \mathrm { z } } { \partial \mathrm { t } }$
C) $\frac { \partial w } { \partial t } = \frac { \partial w } { \partial x } + \frac { \partial w } { \partial y } + \frac { \partial w } { \partial z }$
D) $\frac { \partial \mathrm { w } } { \partial \mathrm { t } } = \frac { \partial \mathrm { w } } { \partial \mathrm { x } } \frac { \partial \mathrm { x } } { \partial \mathrm { t } } + \frac { \partial \mathrm { w } } { \partial \mathrm { y } } \frac { \partial \mathrm { y } } { \partial \mathrm { t } } + \frac { \partial \mathrm { w } } { \partial \mathrm { z } } \frac { \partial \mathrm { z } } { \partial \mathrm { t } }$

Question

Quizplus · Accepted Answer

The chain rule formula for a function with multiple variables can be written as follows: $\frac{\partial w}{\partial t} = \frac{\partial w}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial w}{\partial y}\frac{\partial y}{\partial t} + \frac{\partial w}{\partial z}\frac{\partial z}{\partial t}$. Therefore, choice D is the correct formula for this problem.

Write a Chain Rule Formula for the Following Derivative A)