Use implicit differentiation to find the specified derivative at the given point.
-Find $\frac { \partial y } { \partial x }$ at the point $( 2,3,2 )$ for $3 x ^ { 2 } + 6 \ln x z - 4 y z ^ { 2 } + 4 e ^ { z } = 0$.
A) $\frac { 9 } { 16 }$
B) $- \frac { 15 } { 16 }$
C) $\frac { 15 } { 8 }$
D) $\frac { 15 } { 16 }$

Question

Quizplus · Accepted Answer

Implicit differentiation gives$$6x+6\frac{z}{x}-4yz^{2}-8yzz^{'}+4e^{z}z^{'}=0$$At the given point, $x=2, y=3, z=2$, so$$6(2)+6\frac{2}{2}-4(3)(2)^{2}-8(3)(2)(2^{'})+4e^{2}(2^{'})=0$$$$12+6-48-96+16e^{2}(2^{'})=0$$$$-122+16e^{2}(2^{'})=0$$$$2^{'}=\frac{122}{16e^{2}}=\frac{15}{8e^{2}}$$$$\frac{\partial y}{\partial x}=\frac{-6x-6\frac{z}{x}+4yz^{2}}{4yz^{2}+8yzz^{'}}=\frac{-6(2)-6\frac{2}{2}+4(3)(2)^{2}}{4(3)(2)^{2}+8(3)(2)(\frac{15}{8e^{2}})}=\frac{-12-6+48}{24+90}=\frac{15}{16}$$

Use Implicit Differentiation to Find the Specified Derivative at the Given