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

Question

Quizplus · Accepted Answer

The chain rule states that for a function u of variables p and q, and variables p and q being functions of variables x, y, and z, the derivative of u with respect to x is equal to the partial derivative of u with respect to p times the partial derivative of p with respect to x, plus the partial derivative of u with respect to q times the partial derivative of q with respect to x. Therefore, A is the correct formula. B is incorrect because it has the partial derivative of x with respect to p and q instead of the other way around. C is incorrect because it only includes the partial derivative of q with respect to x and ignores the contribution of p. D is incorrect because it includes the partial derivative of q with respect to y and z, which is not part of the given function.

Write a Chain Rule Formula for the Following Derivative A)