Find the requested partial derivative.
-$( \partial z / \partial x ) _ { y }$ at $( x , y , z ) = ( 1,1,1 )$ if $z ^ { 3 } + 6 x y z = 7$
A) $- \frac { 1 } { 2 }$
B) $- \frac { 4 } { 3 }$
C) $- \frac { 2 } { 5 }$
D) $- \frac { 2 } { 3 }$

Question

Quizplus · Accepted Answer

Using implicit differentiation, we have:

$3z^2 \frac{\partial z}{\partial x}  + 6y z + 6x\frac{\partial z}{\partial x} = 0$

Isolating $\frac{\partial z}{\partial x}$, we get:

$\frac{\partial z}{\partial x} = -\frac{2yz}{3z^2 + 6x} = -\frac{2}{3}$ at $(1,1,1)$

Therefore, the correct answer is D.

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

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