Hello qasdc,
I think one has to be careful here. When it comes to operators, there can always be some problem hiding there. Consider function [itex]H(p,q) = q + pq  qp[/itex]. I'm not familiar with this kind of differentiation, but your definition seems to give
[tex]
\frac{\partial H}{\partial q} = 1 + p  p = 1.
[/tex]
But clearly [itex] pq qp \neq f(p)[/itex].
