# Anti-Commutator of derivatives of Grassmann variables

How do I evaluate the anti-commutator of $$\frac{\partial}{\partial\chi}$$ and $$\frac{\partial}{\partial\eta}$$ when both $$\chi$$ and $$\eta$$ are Grassmann variables?

• ... act with them on an arbitrary function and see what happens? – AccidentalFourierTransform May 5 at 17:32

More generally, one may show that derivatives of supernumbers $$z$$ and $$w$$, with definite Grassmann parities $$|z|$$ and $$|w|$$, resectively, supercommute: $$\frac{\partial}{\partial z}\frac{\partial}{\partial w}~=~(-1)^{|z||w|}\frac{\partial}{\partial w}\frac{\partial}{\partial z} .$$