The complex version of the chain rule I want to prove the following equality:
\begin{eqnarray}
\frac{\partial}{\partial z} (g \circ f) = (\frac{\partial g}{\partial z} \frac{\partial f}{\partial z}) + (\frac{\partial g}{\partial \bar{z}} \frac{\partial \bar{f}}{\partial z})
\end{eqnarray}
So I decide to do the following:
\begin{eqnarray}
\frac{\partial}{\partial z} (g \circ f) = \frac{1}{2}[(\frac{\partial g}{\partial x} \circ f)(\frac{\partial f}{\partial x}) + \frac{1}{i}(\frac{\partial g}{\partial y} \circ f)(\frac{\partial f}{\partial y})]
\end{eqnarray}
but the thing is that I am doing something wrong here since I don't get any conjugate function and any derivative with respect to $\bar{z}$ so Can someone help me to see where I am wrong and fix it please?
In fact I don't see what to do next, so I appreciate your help.
Thanks a lot in advance.

Edition:

What I've got so far is the following:
$$\frac{1}{2}[(\frac{\partial g}{\partial x} \circ f + \frac{\partial g}{\partial y} \circ f)\frac{\partial f}{\partial z} ]$$
but I'm still stuck.

You do some mistakes. Note that $$\frac{\partial}{\partial x} (g \circ f) = (\frac{\partial g}{\partial x} \circ f)(\operatorname{Re}\frac{\partial f}{\partial x})+(\frac{\partial g}{\partial y} \circ f)(\operatorname{Im}\frac{\partial f}{\partial x})$$ and $$\frac{\partial}{\partial y} (g \circ f) = (\frac{\partial g}{\partial x} \circ f)(\operatorname{Re}\frac{\partial f}{\partial y})+(\frac{\partial g}{\partial y} \circ f)(\operatorname{Im}\frac{\partial f}{\partial y}).$$ Now you should be able to finish your calculations.
