Find the requested partial derivative.
-$( \partial \mathrm { w } / \partial \mathrm { z } ) _ { \mathrm { x } }$ at $( \mathrm { x } , \mathrm { y } , \mathrm { z } , \mathrm { w } ) = ( 1,2,9,302 )$ if $\mathrm { w } = \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } + 12 \mathrm { xyz }$ and $\mathrm { z } = \mathrm { x } ^ { 3 } + \mathrm { y } ^ { 3 }$
A) $\frac { 308 } { 3 }$
B) $\frac { 154 } { 3 }$
C) 154
D) 77

Question

Quizplus · Accepted Answer

To find $( \partial \mathrm { w } / \partial \mathrm { z } ) _ { \mathrm { x }}$, we first differentiate $w$ with respect to $z$, treating $x$ as a constant (since we're considering the partial derivative with respect to $z$ at constant $x$). The derivative of $w = x^2 + y^2 + z^2 + 12xyz$ with respect to $z$ is $2z + 12xy$. However, since $z = x^3 + y^3$, we substitute $z = 9$ (given), $x = 1$, and $y = 2$ into the derivative to get $2(9) + 12(1)(2) = 18 + 24 = 42$. This calculation seems straightforward, but the correct approach involves recognizing that the derivative of $w$ with respect to $z$ directly is $2z + 12xy$, and then substituting the values of $x$, $y$, and $z$ to find the specific numerical value at the given point. Given the options provided, none directly match the expected calculation, suggesting a need to reevaluate the explanation or the interpretation of the question's requirements. The correct answer should be based on the derivative $2z + 12xy$ and substituting the given values. However, the provided options and the initial explanation indicate a misunderstanding or misinterpretation of the question's requirements. The correct process involves differentiating $w$ with respect to $z$, considering $z$ as a function of $x$ and $y$ if necessary, but the specific numerical answer provided in the options does not directly follow from the explanation given.

Find the Requested Partial Derivative (∂w/∂z)x( \partial \mathrm { w } / \partial \mathrm { z } ) _ { \mathrm { x } }(∂w/∂z)x​

Find the Requested Partial Derivative $( \partial \mathrm { w } / \partial \mathrm { z } ) _ { \mathrm { x } }$