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

Question

Quizplus · Accepted Answer

This is the chain rule for partial differentiation in two variables. It takes into account the fact that z depends on both r and s, which in turn depend on t. The first term in the sum considers the effect of changes in r on z, while the second term considers the effect of changes in s on z. Each of these effects is multiplied by the corresponding rate of change between the variables.

Write a Chain Rule Formula for the Following Derivative A) ∂z∂t=∂z∂rdrds\frac { \partial z } { \partial t } = \frac { \partial z } { \partial r } \frac { d r } { d s }∂t∂z​=∂r∂z​dsdr​

Write a Chain Rule Formula for the Following Derivative A) $\frac { \partial z } { \partial t } = \frac { \partial z } { \partial r } \frac { d r } { d s }$