# Using FullSimplify to simply expressions involving differential operators

I am trying to simplify some long formulas involving differential operators in $$n$$ variables. For example, I want to simplify $$\left(\frac{\partial}{\partial x}-\frac{\partial}{\partial y}\right) \circ \left(2\frac{\partial}{\partial x}+\frac{\partial}{\partial y}\right) + \left( \frac{\partial^2}{\partial^2 x} - \frac{\partial}{\partial x \partial y}\right)$$ so that the final expression is a linear combination of first and second partial derivatives. For this I used the following code,

FullSimplify[
Composition[
D[f[x, y], x] - D[f[x, y], y],
2*D[f, x] + D[f, y]] + (D[f[x, y], {x, 2}] - D[f[x, y], x, y])];


However, this doesn't seem to be working. Mathematica is giving me exactly the same expression back. Am I doing something wrong?