Write a chain rule formula for the following derivative.
-$$\frac { \partial \mathrm { w } } { \partial \mathrm { t } } \text { for } \mathrm { w } = \mathrm { f } ( \mathrm { p } , \mathrm { q } , \mathrm { r } ) ; \mathrm { p } = \mathrm { g } ( \mathrm { t } ) , \mathrm { q } = \mathrm { h } ( \mathrm { t } ) , \mathrm { r } = \mathrm { k } ( \mathrm { t } )$$
A) $\frac { \partial \mathrm { w } } { \partial \mathrm { t } } = \frac { \mathrm { dp } } { \mathrm { dt } } + \frac { \mathrm { dq } } { \mathrm { dt } } + \frac { \mathrm { dr } } { \mathrm { dt } }$
B) $\frac { \partial \mathrm { w } } { \partial \mathrm { t } } = \frac { \partial \mathrm { w } } { \partial \mathrm { p } } \frac { \mathrm { dt } } { \mathrm { dp } } + \frac { \partial \mathrm { w } } { \partial \mathrm { q } } \frac { \mathrm { dt } } { \mathrm { dq } } + \frac { \partial \mathrm { w } } { \partial \mathrm { r } } \frac { \mathrm { dt } } { \mathrm { dr } }$
C) $\frac { \partial \mathrm { w } } { \partial \mathrm { t } } = \frac { \partial \mathrm { w } } { \partial \mathrm { p } } + \frac { \partial \mathrm { w } } { \partial \mathrm { q } } + \frac { \partial \mathrm { w } } { \partial \mathrm { r } }$
D) $\frac { \partial \mathrm { w } } { \partial \mathrm { t } } = \frac { \partial \mathrm { w } } { \partial \mathrm { p } } \frac { \mathrm { dp } } { \mathrm { dt } } + \frac { \partial \mathrm { w } } { \partial \mathrm { q } } \frac { \mathrm { dq } } { \mathrm { dt } } + \frac { \partial \mathrm { w } } { \partial \mathrm { r } } \frac { \mathrm { dr } } { \mathrm { dt } }$

Question

Quizplus · Accepted Answer

The correct formula applies the chain rule for partial derivatives, accounting for how w changes with respect to t through the intermediary variables p, q, and r, each of which is a function of t.

Write a Chain Rule Formula for the Following Derivative A)