Provide an Appropriate Response $\mathrm { P } _ { 0 }$

Multiple Choice

Provide an appropriate response.
-Find the direction in which the function is increasing most rapidly at the point $\mathrm { P } _ { 0 }$ .
$$f ( x , y ) = x e ^ { y } - \ln ( x ) , P _ { 0 } ( - 3,0 )$$

A) $\left( - \frac { 9 } { \sqrt { 97 } } \right) i + \left( \frac { 4 } { \sqrt { 97 } } \right) j$
B) $\left( - \frac { 4 } { \sqrt { 97 } } \right) \mathrm { i } - \left( \frac { 9 } { \sqrt { 97 } } \right) \mathrm { j }$
C) $\left( - \frac { 4 } { \sqrt { 97 } } \right) \mathrm { i }$
D) $\left( \frac { 4 } { \sqrt { 97 } } \right) \mathrm { i } - \left( \frac { 9 } { \sqrt { 97 } } \right) \mathrm { j }$