Answer
$$\nabla g\left( {x,y} \right) = {e^{ - x}}\left( { - y{\bf{i}} + {\bf{j}}} \right),{\text{ }}\sqrt {26} $$
Work Step by Step
$$\eqalign{
& g\left( {x,y} \right) = y{e^{ - x}},{\text{ point }}\left( {0,5} \right) \cr
& {\text{Find the partial derivatives }}{g_x}\left( {x,y} \right){\text{ and }}{g_y}\left( {x,y} \right) \cr
& {g_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {y{e^{ - x}}} \right] \cr
& {g_x}\left( {x,y} \right) = - y{e^{ - x}} \cr
& and \cr
& {g_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {y{e^{ - x}}} \right] \cr
& {g_y}\left( {x,y} \right) = {e^{ - x}} \cr
& {\text{Calculate }}\nabla g\left( {x,y} \right) \cr
& \nabla g\left( {x,y} \right) = {g_x}\left( {x,y} \right){\bf{i}} + {g_y}\left( {x,y} \right){\bf{j}} \cr
& \nabla g\left( {x,y} \right) = - y{e^{ - x}}{\bf{i}} + {e^{ - x}}{\bf{j}} \cr
& \nabla g\left( {x,y} \right) = {e^{ - x}}\left( { - y{\bf{i}} + {\bf{j}}} \right) \cr
& \cr
& {\text{Evaluate }}\nabla g\left( {0,5} \right) \cr
& \nabla g\left( {0,5} \right) = - \left( 5 \right){e^{ - 0}}{\bf{i}} + {e^{ - 0}}{\bf{j}} \cr
& \nabla g\left( {0,5} \right) = - 5{\bf{i}} + {\bf{j}} \cr
& \cr
& {\text{The maximum value of }}{D_{\bf{u}}}g\left( {x,y} \right){\text{ is }}\left\| {\nabla g\left( {x,y} \right)} \right\| \cr
& \left\| {\nabla g\left( {0,5} \right)} \right\| = \left\| { - 5{\bf{i}} + {\bf{j}}} \right\| \cr
& \left\| {\nabla g\left( {0,5} \right)} \right\| = \sqrt {25 + 1} \cr
& \left\| {\nabla g\left( {0,5} \right)} \right\| = \sqrt {26} \cr} $$
