Answer
$\nabla \phi (x, y)=\lt \dfrac{-x}{(x^2+y^2+z^2)^{3/2}}, \dfrac{-y}{(x^2+y^2+z^2)^{3/2}}, \dfrac{-z}{(x^2+y^2+z^2)^{3/2}} \gt$
Work Step by Step
Our aim is to compute the the gradient vector field vector.
Here, we have $\phi (x, y)=(x^2+y^2+z^2)^{-1/2}$
Now, $\nabla \phi (x, y)=\lt \dfrac{\partial \phi (x, y)}{\partial x} , \dfrac{\partial \phi (x, y)}{\partial y} , \dfrac{\partial \phi (x, y)}{\partial z} \gt $
or, $\nabla \phi (x, y)=\lt \dfrac{\partial (x^2+y^2+z^2)^{-1/2})}{\partial x} , \dfrac{\partial (x^2+y^2+z^2)^{-1/2}))}{\partial y}, \dfrac{\partial (x^2+y^2+z^2)^{-1/2})}{\partial z} \gt \\= \lt - \dfrac{(2x+0+0)}{2} (x^2+y^2+z^2)^{-3/2} , - \dfrac{(0+2y++0)}{2} (x^2+y^2+z^2)^{-3/2}, - \dfrac{(0+0+2z)}{2} (x^2+y^2+z^2)^{-3/2} \gt $
Thus, we have $\nabla \phi (x, y)=\lt \dfrac{-x}{(x^2+y^2+z^2)^{3/2}}, \dfrac{-y}{(x^2+y^2+z^2)^{3/2}}, \dfrac{-z}{(x^2+y^2+z^2)^{3/2}} \gt$
