$$\left.{\partial X\over \partial Y}\right|_Z=-\frac{\left.{\partial Z\over \partial Y}\right|_X}{ \left.{\partial Z\over \partial X}\right|_Y}$$
The above is an identity frequently used in thermodynamics when $X$,$Y$, and $Z$ satisfy $\phi(X, Y, Z) = const$ for some function $\phi$. My thermodynamics book drives the identity like this: Let's assume the constaint can be solved for $Z$, so we have $Z(X, Y, \phi)=const$. The differential of $Z$ is written as $$0 = dZ = \left.\frac{\partial Z}{\partial X}\right|_{Y,\phi} dx + \left.\frac{\partial Z}{\partial Y}\right|_{X,\phi} dy + \left.\frac{\partial Z}{\partial \phi}\right|_{X,Y} d\phi$$ Now we set $d\phi=0$, and devide the above equation by $dy$, and identify $dx/dy$ as $\left.\frac{\partial X}{\partial Y}\right|_{Z,\phi}$. Finally we drop the $\phi$ in the subscripts of partial derivatives (this is just a convention). Then we get the identity. Is this acceptable as a mathematical proof? If not, I'd like to know how to prove it. And I'd like to know whether there is a rigorous way to justify $dy/dx=\left.\frac{\partial X}{\partial Y}\right|_{Z,\phi}$ in terms of "differential forms" or in terms of "differentials in calculus".