Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be  a  polynomial vector field on $\mathbb{R}^{2}$.  consider the  following (Moyal) derivation on $\mathbb{C}[x,y]$:

$\tilde{D}_{X}(f)=f_{x}*P+f_{y}*Q$  where $*$ is the Moyal product. This is a noncommutativized version of the standard derivation $D_{X}(f)=f_{x}P+f_{y}Q$:

I have two questions:

1) Motivating by  the Leibniz formula for $D_{X}$, I  ask that what is a natural  formula  for $\tilde{D}_{X}(f*g)=?$

2)Can the number of  limit  cycles of $X$ be  bounded by the  codimension of the range of $\tilde{D}_{X}$?  


  A (commutative) motivation  for  the  second  question is the question in the following  post;

http://mathoverflow.net/questions/164059/codimension-of-the-range-of-certain-linear-operators