I was reading Narasimhan's Complex Analysis book, and in the part where he explains the maximum modulus principle, he introduces two operators $\Delta u=\frac{\partial^2 u}{\partial^2 x}+\frac{\partial^2 u}{\partial^2 y}$ and $\Delta^c u=\frac{\partial^2 u}{\partial z \partial \bar{z}}$ (which is actually just $\Delta^c=\frac{1}{4}\Delta$, really) and goes like...
If $u(z)=|f(z)|^2=f(z)\overline{f(z)}$, we have $$\Delta u=4\Delta^c u=4\frac{\partial f}{\partial z}\cdot\frac{\partial \bar{f}}{\partial \bar{z}}=4|f'(z)|^2 \geq 0.$$
and this is essentially the whole content of the proof.
Now, my question is, how does one get the expression $\frac{\partial f}{\partial z}\cdot\frac{\partial \bar{f}}{\partial \bar{z}}$ so quickly from $\Delta^c u$?
For me, the only way to prove this is to plug in all the $\partial x$'s and $\partial y$'s according to the definitions of $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \bar{z}}$ and see where the tedious calculation leads. But since the proof is so short, I guess there is some other way to see this in a very simple manner. So... How??