Proof for $1={\left( \frac{\partial x }{\partial y} \right)_{\!z}}{\left( \frac{\partial y }{\partial x} \right)_{\!z}}$

Question

I was solving some thermodynamics problems and encountered these two equations:

$$1={\left( \frac{\partial x }{\partial y} \right)_{\!z}}{\left( \frac{\partial y }{\partial x} \right)_{\!z}}$$

and

$${\left( \frac{\partial x }{\partial y} \right)_{\!z}}{\left( \frac{\partial y }{\partial z} \right)_{\!x}}=-{\left( \frac{\partial x }{\partial z} \right)_{\!y}}$$

Are there proofs for these(that my puny brain could handle)? And could someone show an example function f(x,y,z) of these showing to be true?

Also, why does it seem like we can treat the partials as fractions in the first equation, but not in the second? When am I allowed to treat them as fractions?

Answer 1

HINT

Research Maxwell Relations, which explain exactly how 3 variables correlate given the symmetry of their second order derivatives, in this case $(x, y, z)$.