Show That At $\left( \frac { 1 } { \sqrt { 2 } } , 0 \right)$

Essay

Show that at $\left( \frac { 1 } { \sqrt { 2 } } , 0 \right)$ , the equation $x ^ { 2 } + \frac { 1 } { 2 } y ^ { 2 } + \frac { 1 } { 3 } z ^ { 2 } = 1$ defines $Z$ implicitly as a function of $x$ and $y$ , and then compute $\frac { \delta z } { \delta x } \left( \frac { 1 } { \sqrt { 2 } } , 0 , \sqrt { \frac { 3 } { 2 } } \right)$ and $\frac { \delta z } { \delta y } \left( \frac { 1 } { \sqrt { 2 } } , 0 , \sqrt { \frac { 3 } { 2 } } \right)$ .