Let $$ f(x_1,x_2) = \frac{1}{\sqrt{x_1^2+x_2^2+1}} \left(x_1,x_2,1 \right) $$
For $i = 1,2$ we have
$$ \partial_{x_i} f = \frac{1}{\sqrt{x_1^2+x_2^2+1}}\left(\partial_{x_i}x_1,\partial_{x_i}x_2,0 \right) +\frac{-x_i}{\left(x_1^2+x_2^2+1\right)^{3/2}}(x_1,x_2,1) = \frac{1}{\left(x_1^2+x_2^2+1\right)^{3/2}} \left[\left(x_1^2+x_2^2+1\right)\left(\partial_{x_i}x_1,\partial_{x_i}x_2,0\right) - x_i(x_1,x_2,1) \right] $$
The gradient of such function, after some computation I get the following matrix
$$ \nabla f = \frac{1}{\left(x_1^2+x_2^2+1\right)^{3/2}} \begin{pmatrix} x_2^2+1 & - x_1x_2 & -x_1 \\ -x_1x_2 & x_1^2+1 & -x_2 \end{pmatrix} $$
Is it correct?