  We study partial derivatives on the product of two metric measure structures,
in particular in connection with calculus via modules as proposed by the first
named author.
  Our main results are 1) The extension to this non-smooth framework of
Schwarz's theorem about symmetry of mixed second derivatives, 2) a quite
complete set of results relating the property $f\in W^{2,2}(\X\times\Y)$ on one
side with that of $f(\cdot,y)\in W^{2,2}(\X)$ and $f(x,\cdot)\in W^{2,2}(\Y)$
for a.e.\ $y,x$ respectively on the other. Here $\X,\Y$ are $\RCD$ spaces so
that second order Sobolev spaces are well defined. \end{itemize} These results
are in turn based upon the study of Sobolev regularity, and of the underlying
notion of differential, for a map with values in a Hilbert module: we mainly
apply this notion to the map $x\mapsto\d_\sy f(x,\cdot)$ in order to build,
under the appropriate regularity requirements, its differential $\d_\sx\d_\sy
f$.
