Answer
$$\eqalign{
& {w_x} = \frac{x}{{\sqrt {{x^2} - {y^2} - {z^2}} }} \cr
& {w_y} = - \frac{y}{{\sqrt {{x^2} - {y^2} - {z^2}} }} \cr
& {w_z} = - \frac{z}{{\sqrt {{x^2} - {y^2} - {z^2}} }} \cr} $$
Work Step by Step
$$\eqalign{
& w = \sqrt {{x^2} - {y^2} - {z^2}} \cr
& {\text{Calculate }}{w_x}{\text{ treating }}y{\text{ and }}z{\text{ as constants}} \cr
& {w_x} = \frac{\partial }{{\partial x}}\left[ {\sqrt {{x^2} - {y^2} - {z^2}} } \right] \cr
& {w_x} = \frac{{\frac{\partial }{{\partial x}}\left[ {{x^2} - {y^2} - {z^2}} \right]}}{{2\sqrt {{x^2} - {y^2} - {z^2}} }} \cr
& {w_x} = \frac{{2x}}{{2\sqrt {{x^2} - {y^2} - {z^2}} }} \cr
& {w_x} = \frac{x}{{\sqrt {{x^2} - {y^2} - {z^2}} }} \cr
& {\text{Calculate }}{w_y}{\text{ treating }}x{\text{ and }}z{\text{ as constants}} \cr
& {w_y} = \frac{\partial }{{\partial y}}\left[ {\sqrt {{x^2} - {y^2} - {z^2}} } \right] \cr
& {w_y} = \frac{{\frac{\partial }{{\partial y}}\left[ {{x^2} - {y^2} - {z^2}} \right]}}{{2\sqrt {{x^2} - {y^2} - {z^2}} }} \cr
& {w_y} = \frac{{ - 2y}}{{2\sqrt {{x^2} - {y^2} - {z^2}} }} \cr
& {w_y} = - \frac{y}{{\sqrt {{x^2} - {y^2} - {z^2}} }} \cr
& {\text{Calculate }}{w_z}{\text{ treating }}x{\text{ and }}y{\text{ as constants}} \cr
& {w_z} = \frac{\partial }{{\partial z}}\left[ {\sqrt {{x^2} - {y^2} - {z^2}} } \right] \cr
& {w_z} = \frac{{\frac{\partial }{{\partial y}}\left[ {{x^2} - {y^2} - {z^2}} \right]}}{{2\sqrt {{x^2} - {y^2} - {z^2}} }} \cr
& {w_z} = \frac{{ - 2z}}{{2\sqrt {{x^2} - {y^2} - {z^2}} }} \cr
& {w_z} = - \frac{z}{{\sqrt {{x^2} - {y^2} - {z^2}} }} \cr} $$
