TITLE: Complex chain rule for complex valued functions
QUESTION [12 upvotes]: Let $f=f(z)$ and $g=g(w)$ be two complex valued functions which are differentiable in the real sense, $h(z)=g(f(z))$. Prove the complex chain rule. 
All partial derivatives:
$$ 
\frac{\partial h}{\partial z} = \frac{\partial g}{\partial w}\frac{\partial f}{\partial z} + \frac{\partial g}{\partial \bar w}\frac{\partial \bar f}{\partial z}
$$
and
$$
\frac{\partial h}{\partial \bar z} = \frac{\partial g}{\partial w}\frac{\partial f}{\partial \bar z} + \frac{\partial g}{\partial \bar w}\frac{\partial\bar f}{\partial \bar z}
$$
Are we supposed to arrive at this through Cauchy-Riemann?

REPLY [14 votes]: This has nothing to do with Cauchy-Riemann's equations. Use the regular chain rule (for functions on $\mathbb{R}^2$) and the definition of the Wirtinger derivatives:
$$
\frac{\partial}{\partial z} = \frac12 \left( \frac{\partial}{\partial x}  - i\,\frac{\partial}{\partial y}  \right)
\qquad\text{and}\qquad
\frac{\partial}{\partial \bar z} = \frac12 \left( \frac{\partial}{\partial x}  + i\,\frac{\partial}{\partial y}  \right)
$$
It all boils down to a fairly long and tedious algebraic manipulation
(See also: Wikipedia)