Answer
$0$
Work Step by Step
Given: $g(x,y,z)=\sqrt{1+xz}+\sqrt {1-xy}$
Differentiating $\sqrt{1+xz}+\sqrt {1-xy}$ partially with respect to $z$, keeping $x$ and $y$ constant, we get:
$g_z=\dfrac{∂[\sqrt{1+xz}+\sqrt {1-xy}]}{∂z}=\dfrac{x}{2\sqrt{1+xz}}$
Differentiating the above equation partially with respect to $y$, keeping $z$ and $z$ constant, we get:
$g_zy=\dfrac{∂[\dfrac{x}{2\sqrt{1+xz}}]}{∂y}=0$
and
$g_{zyx}=0$
Thus, the partial derivatives are continuous on their domains and as per Clairaut's Theorem, this implies that all the third order derivatives become $0$.
Hence, $g_{zxy}=g_{xyz}=0$
