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?