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 $\mathrm { P } _ { 0 }$ . $f ( x , y , z ) = x y - \ln ( z ) , P _ { 0 } ( 1,2,2 )$

A) $\frac { 1 } { 21 } ( 4 \mathrm { i } + 2 \mathrm { j } - \mathrm { k } )$
B) $\frac { 1 } { \sqrt { 21 } } ( 4 i - 2 j + k )$
C) $\frac { 21 } { \sqrt { 21 } } ( 4 i + 2 j - k )$
D) $\frac { 1 } { \sqrt { 21 } } ( 4 i + 2 j - k )$