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