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

Question

Quizplus · Accepted Answer

The chain rule formula for the given derivative is:
$$\frac{\partial w}{\partial t} = \frac{\partial w}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial w}{\partial y}\frac{\partial y}{\partial t} + \frac{\partial w}{\partial z}\frac{\partial z}{\partial t}$$
Since $z=k(s)$ and $\frac{\partial z}{\partial t}=0$, the last term drops out and we are left with:
$$\frac{\partial w}{\partial t} = \frac{\partial w}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial w}{\partial y}\frac{\partial y}{\partial t}$$
This matches option B. Option A is incorrect because it is missing the term $\frac{\partial w}{\partial z}\frac{\partial z}{\partial t}$ which is 0 in this case. Option C is incorrect because it only considers the partial derivative with respect to $x$ and ignores $y$ and $z$. Option D is incorrect because it does not take into account the dependence of $w$ on $y$ and $z$.

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

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