Answer
$$\nabla w = {e^{8y}}\left( {\sin 6z{\bf{i}} + 8x\sin 6z{\bf{j}} + 6x\cos 6z{\bf{k}}} \right)$$
Work Step by Step
$$\eqalign{
& w = x{e^{8y}}\sin 6z \cr
& \cr
& {\text{calculate the partial derivatives }}{w_x}{\text{,}}\,\,{w_y}{\text{, and }}{w_z} \cr
& {w_x} = \frac{\partial }{{\partial x}}\left[ {x{e^{8y}}\sin 6z} \right] \cr
& {\text{treat }}y{\text{ and }}z{\text{ as constants}} \cr
& {w_x} = {e^{8y}}\sin 6z \cr
& \cr
& {w_y} = \frac{\partial }{{\partial y}}\left[ {x{e^{8y}}\sin 6z} \right] \cr
& {\text{treat }}x{\text{ and }}z{\text{ as constants}} \cr
& {w_y} = x\sin 6z\left( {8{e^{8y}}} \right) \cr
& {w_y} = 8x{e^{8y}}\sin 6z \cr
& \cr
& {w_z} = \frac{\partial }{{\partial z}}\left[ {x{e^{8y}}\sin 6z} \right] \cr
& {\text{treat }}x{\text{ and }}y{\text{ as constants}} \cr
& {w_z} = x{e^{8y}}\left( {6\cos 6z} \right) \cr
& {w_z} = 6x{e^{8y}}\cos 6z \cr
& \cr
& {\text{The gradient of the function }}w\left( {x,y,z} \right){\text{ is defined by }}\left( {{\text{see page 963}}} \right) \cr
& \nabla w = {z_x}{\bf{i}} + {z_y}{\bf{j}} + {z_z}{\bf{k}} \cr
& {\text{substituting the partial derivatives, we obtain}} \cr
& \nabla w = {e^{8y}}\sin 6z{\bf{i}} + 8x{e^{8y}}\sin 6z{\bf{j}} + 6x{e^{8y}}\cos 6z{\bf{k}} \cr
& {\text{factoring}} \cr
& \nabla w = {e^{8y}}\left( {\sin 6z{\bf{i}} + 8x\sin 6z{\bf{j}} + 6x\cos 6z{\bf{k}}} \right) \cr} $$
