Write a chain rule formula for the following derivative.
-$\frac { \partial \mathrm { u } } { \partial \mathrm { t } }$ for $\mathrm { u } = \mathrm { f } ( \mathrm { v } ) ; \mathrm { v } = \mathrm { h } ( \mathrm { s } , \mathrm { t } )$
A) $\frac { \partial \mathrm { u } } { \partial \mathrm { t } } = \frac { \mathrm { dv } } { \mathrm { du } } \frac { \partial \mathrm { u } } { \partial \mathrm { t } }$
B) $\frac { \partial \mathrm { u } } { \partial \mathrm { t } } = \frac { \mathrm { du } } { \mathrm { dv } } \frac { \partial \mathrm { v } } { \partial \mathrm { t } }$
C) $\frac { \partial \mathrm { u } } { \partial \mathrm { t } } = \frac { \mathrm { du } } { \mathrm { dv } }$
D) $\frac { \partial \mathrm { u } } { \partial \mathrm { t } } = \frac { \partial \mathrm { v } } { \partial \mathrm { t } }$

Question

Quizplus · Accepted Answer

The chain rule formula for this derivative is given by:
$$\frac{\partial u}{\partial t} = \frac{\partial u}{\partial v} \frac{\partial v}{\partial t}$$
Substituting the given expressions for $u$ and $v$ gives:
$$\frac{\partial f(v)}{\partial t} = \frac{\partial f(v)}{\partial v} \frac{\partial h(s,t)}{\partial t}$$
which simplifies to:
$$\frac{\partial u}{\partial t} = \frac{\partial f}{\partial v} \frac{\partial h}{\partial t}$$
Therefore, the correct choice is B.

Write a Chain Rule Formula for the Following Derivative ∂u∂t\frac { \partial \mathrm { u } } { \partial \mathrm { t } }∂t∂u​

Write a Chain Rule Formula for the Following Derivative $\frac { \partial \mathrm { u } } { \partial \mathrm { t } }$