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

Question

Quizplus · Accepted Answer

This problem requires the chain rule formula for partial derivatives. In this case, we have three variables: x, y, and z, all of which depend on t. Using the chain rule, we can express the derivative of w with respect to t in terms of the partial derivatives of w with respect to x, y, and z, and the derivatives of x, y, and z with respect to t. The correct formula is:
$$\frac { \partial w } { \partial t } = \frac { \partial w } { \partial x } \frac { \partial x } { \partial t } + \frac { \partial w } { \partial y } \frac { \partial y } { \partial t } + \frac { \partial w } { \partial z } \frac { \partial z } { \partial t }$$
Using the given relationships between the variables, we can substitute the expressions for x, y, and z in terms of r, s, and t into this formula to get:
$$\frac { \partial w } { \partial t } = \frac { \partial w } { \partial x } \left( \frac { \partial g } { \partial t } \right) + \frac { \partial w } { \partial y } \left( \frac { \partial h } { \partial t } \right) + \frac { \partial w } { \partial z } \left( \frac { \partial k } { \partial t } \right)$$
which simplifies to the given answer choice A.

Write a Chain Rule Formula for the Following Derivative ∂w∂t for w=f(x,y,z);x=g(r,s),y=h(t),z=k(r,s,t)\frac { \partial w } { \partial t } \text { for } w = f ( x , y , z ) ; x = g ( r , s ) , y = h ( t ) , z = k ( r , s , t )∂t∂w​ for w=f(x,y,z);x=g(r,s),y=h(t),z=k(r,s,t)

Write a Chain Rule Formula for the Following Derivative $\frac { \partial w } { \partial t } \text { for } w = f ( x , y , z ) ; x = g ( r , s ) , y = h ( t ) , z = k ( r , s , t )$