Partial derivative mnemonic? The following partial derivative mnemonic (with Jacobians) seems to work well in thermodynamics: $\frac{\partial(A,B)}{\partial(C,B)}=\left(\frac{\partial A}{\partial C}\right)_B$ $\partial(A,B)=-\partial(B,A)$ Now it seems I can even treat the individual parts as single elements and get correct results without memorizing all sorts of partial derivative rules. For thermodynamics in particular all I need to know is $\partial(p,V)=\partial(S,T)$. Can this mnemonic have a solid foundation in mathematics as it seems to work well?

The first part of what you call a mnemonic is not a mnemonic but a fact:

$$ \def\jacob#1#2{\frac{\partial(#1)}{\partial(#2)}} \def\pderiv#1#2#3{\left(\frac{\partial #1}{\partial #2}\right)_{#3}} \jacob{A,B}{C,B} = \left| \begin{array}{cc} \pderiv ACB & \pderiv ABC\\\ \pderiv BCB & \pderiv BBC \end{array} \right| = \left| \begin{array}{cc} \pderiv ACB & \pderiv ABC\\\ 0 & 1 \end{array} \right| =\pderiv ACB\;. $$

The second part is only mnemonic in that the numerator and denominator of the Jacobian don't make sense individually, but its foundation in the antisymmetry of determinants is clear.

Then you just need the chain rule

$$ \jacob{x_1,x_2}{y_1,y_2}\jacob{y_1,y_2}{z_1,z_2}=\jacob{x_1,x_2}{z_1,z_2} $$

to justify treating Jacobians as fractions.