Answer
$$\nabla w = 6{\bf{i}} - 10{\bf{j}} - 8{\bf{k}}$$
Work Step by Step
$$\eqalign{
& w = 3{x^2} - 5{y^2} + 2{z^2},{\text{ }}\left( {1,1, - 2} \right) \cr
& {\text{Calculate the partial derivatives }}{w_x}{\text{, }}{w_y}{\text{ and }}{w_z} \cr
& {w_x} = \frac{\partial }{{\partial x}}\left[ {3{x^2} - 5{y^2} + 2{z^2}} \right] \cr
& {w_x} = 6x \cr
& \cr
& {w_y} = \frac{\partial }{{\partial y}}\left[ {3{x^2} - 5{y^2} + 2{z^2}} \right] \cr
& {w_y} = - 10y \cr
& \cr
& {w_z} = \frac{\partial }{{\partial z}}\left[ {3{x^2} - 5{y^2} + 2{z^2}} \right] \cr
& {w_z} = 4z \cr
& \cr
& {\text{The gradient of }}w{\text{ is}} \cr
& \nabla w = {w_x}{\bf{i}} + {w_y}{\bf{j}} + {w_z}{\bf{k}} \cr
& \nabla w = 6x{\bf{i}} - 10y{\bf{j}} + 4z{\bf{k}} \cr
& {\text{At the point }}\left( {1,1, - 2} \right){\text{ the gradient is}} \cr
& \nabla w = 6\left( 1 \right){\bf{i}} - 10\left( 1 \right){\bf{j}} + 4\left( { - 2} \right){\bf{k}} \cr
& \nabla w = 6{\bf{i}} - 10{\bf{j}} - 8{\bf{k}} \cr} $$
