Answer
$0$
Work Step by Step
Let $g(x, y, z)=g_{1}(x, y, z)+g_{2}(x, y, z)$
$ g_{1}=\sqrt{1+xz}, \qquad$ and, we have
$g_{1(y)}(x, y, z)=0\Rightarrow g_{1(yx)}(x, y, z)=0\Rightarrow g_{1(yxz)}(x, y, z)=0$
$g_{2}(x, y, z) =\sqrt{1-xy}\qquad$ and, we have
$g_{2(z)}(x, y, z)=0\Rightarrow g_{2(zx)}(x, y, z)=0\Rightarrow g_{2(zxy)}(x, y, z)=0$
The partial derivatives are continuous on their domains, so by Clairaut's theorem,
all third order partial derivatives of both $g_{1}$ and $g_{2}$ are zero, and
$g_{xyz} =0+0=0$
