Write a chain rule formula for the following derivative.
-$\frac { \partial \mathrm { u } } { \partial \mathrm { x } }$ for $\mathrm { u } = \mathrm { f } ( \mathrm { r } , \mathrm { s } , \mathrm { t } ) ; \mathrm { r } = \mathrm { g } ( \mathrm { y } ) , \mathrm { s } = \mathrm { h } ( \mathrm { z } ) , \mathrm { t } = \mathrm { k } ( \mathrm { x } , \mathrm { z } )$
A) $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = 0$
B) $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = \frac { \partial \mathrm { u } } { \partial \mathrm { t } }$
C) $\frac { \partial u } { \partial x } = \frac { \partial t } { \partial x }$
D) $\frac { \partial \mathrm { u } } { \partial \mathrm { x } } = \frac { \partial \mathrm { u } } { \partial \mathrm { t } } \frac { \partial \mathrm { t } } { \partial \mathrm { x } }$

Question

Quizplus · Accepted Answer

To use the chain rule, we need to start by differentiating u with respect to t, r, and s. This gives us:
$$\frac{\partial u}{\partial t} = \frac{\partial f}{\partial t} + \frac{\partial f}{\partial r}\frac{\partial r}{\partial t} + \frac{\partial f}{\partial s}\frac{\partial s}{\partial t}$$
$$\frac{\partial u}{\partial r} = \frac{\partial f}{\partial r}$$
$$\frac{\partial u}{\partial s} = \frac{\partial f}{\partial s}$$

Next, we differentiate t with respect to x and we get:
$$\frac{\partial t}{\partial x} = \frac{\partial k}{\partial x} + \frac{\partial k}{\partial z}\frac{\partial z}{\partial x}$$

Finally, using the chain rule, we can differentiate u with respect to x, which gives us:
$$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial t}\frac{\partial t}{\partial x} + \frac{\partial u}{\partial r}\frac{\partial r}{\partial x} + \frac{\partial u}{\partial s}\frac{\partial s}{\partial x}$$
Substituting the expressions we found earlier, we get:
$$\frac{\partial u}{\partial x} = \left(\frac{\partial f}{\partial t} + \frac{\partial f}{\partial r}\frac{\partial r}{\partial t} + \frac{\partial f}{\partial s}\frac{\partial s}{\partial t}\right)\left(\frac{\partial k}{\partial x} + \frac{\partial k}{\partial z}\frac{\partial z}{\partial x}\right) + \frac{\partial f}{\partial r}\frac{\partial r}{\partial x} + \frac{\partial f}{\partial s}\frac{\partial s}{\partial x}$$
Simplifying this expression we get:
$$\frac{\partial u}{\partial x} = \frac{\partial f}{\partial t}\frac{\partial k}{\partial x} + \frac{\partial f}{\partial r}\frac{\partial r}{\partial x}\left(\frac{\partial k}{\partial z}\frac{\partial z}{\partial x}+1\right) + \frac{\partial f}{\partial s}\frac{\partial s}{\partial x}\frac{\partial k}{\partial z}\frac{\partial z}{\partial x}$$
which is equivalent to:
$$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial t}\frac{\partial t}{\partial x} + \frac{\partial u}{\partial r}\frac{\partial r}{\partial x} + \frac{\partial u}{\partial s}\frac{\partial s}{\partial x}$$
Therefore, the correct choice is D.

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

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