Calculating a complex derivative of a polynomial What are the rules for derivatives with respect to $z$ and $\bar{z}$ in polynomials?
For instance, is it justified to calculate the partial derivatives of $f(z,\bar{z})=z^3-2z+\bar{z}-(\overline{z-3i})^4$ as if $z$ and $\bar{z}$ were independent? i.e. $f_z=3z^2-2$ and $f_\bar{z}=1-4(\overline{z-3i})^3$

I would first write $$ f(z,\bar z)=z^3−2z+\bar z−(\bar z+3i)^4 $$
and then treat $z$ and $\bar z$ as independent parameters.
Then I have
$$f_z=3z^2−2$$      $$f_{\bar z}=1−4(\bar z+3i)^3$$
Am I right?
