= Solution
Differentiate the defining function with the other coordinates independent. Put $C=\cosh(x+z)$ for brevity. Then
$$
F_x=yz-C,\qquad F_y=xz,\qquad F_z=xy-C.
$$
The <implicit differentiation> ratios therefore give
$$
\boxed{\left(\frac{\partial x}{\partial y}\right)_z=-\frac{xz}{yz-C},\quad
\left(\frac{\partial x}{\partial z}\right)_y=-\frac{xy-C}{yz-C},\quad
\left(\frac{\partial y}{\partial z}\right)_x=-\frac{xy-C}{xz}.}
$$
Each expression applies on the implicit surface wherever its own denominator is nonzero. The <hyperbolic cosine> appears because the <derivative> of $\sinh(x+z)$ with respect to either $x$ or $z$ is $\cosh(x+z)$.
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