Chain Rule (Complex Analysis) Proposition:
Let $\phi$ be a function that is (real) differentiable in $A$ and let $\gamma: [a,b] \subset \mathbb{R} \rightarrow A$ be differentiable in [a,b], then:
$(\phi \circ \gamma)'(t)=\frac{\partial}{\partial z} \phi(\gamma(t))\gamma'(t)+\frac{\partial}{\partial \overline{z}} \phi(\gamma(t))\overline{\gamma}'(t)$
In the book I am reading this Proposition is just there without a proof. I can't really figure out how to get to this result.
And the second thing is, as I tried to proof the above the result I got was:
$(\phi \circ \gamma)'(t)=$$\phi '(\gamma (t))\gamma '(t)$
Which one of them is the correct one (or are both correct)?
 A: This is a particular case of the chain rule for Wirtinger derivatives, which state that$$\frac\partial{\partial z}(f\circ g)=\left(\frac{\partial f}{\partial z}\circ g\right)\frac{\partial g}{\partial z}+\left(\frac{\partial f}{\partial\overline z}\circ g\right)\frac{\partial\overline g}{\partial z}$$and that$$\frac\partial{\partial\overline z}(f\circ g)=\left(\frac{\partial f}{\partial z}\circ g\right)\frac{\partial g}{\partial\overline z}+\left(\frac{\partial f}{\partial\overline z}\circ g\right)\frac{\partial\overline g}{\partial\overline z}.$$Note that, in order that this makes sence, you only have to assume real-differentiability.
A: This is a consequence of the usual real chain rule
$$(\phi \circ \gamma)^\prime(t)=\phi^\prime (\gamma (t))\gamma^\prime (t)$$
when you use the Cauchy Riemann relations
$$\begin{cases}
\frac{\partial}{\partial z}&= \frac{1}{2}\left(\frac{\partial}{\partial x} -\frac{\partial}{\partial y}\right)\\
\frac{\partial}{\partial \overline z}&= \frac{1}{2}\left(\frac{\partial}{\partial x} +\frac{\partial}{\partial y}\right)
\end{cases}$$
In details, you have
$$\gamma^\prime(t) = \gamma_x^\prime(t) + i \gamma_y^\prime(t)$$ hence
$$\begin{cases}
\gamma_x^\prime(t) &= \frac{1}{2}\left(\gamma^\prime(t)+\overline\gamma^\prime(t)\right)\\
\gamma_y^\prime(t) &= \frac{1}{2i}\left(\gamma^\prime(t)-\overline\gamma^\prime(t)\right)
\end{cases}$$
while
$$\begin{aligned}
(\phi \circ \gamma)^\prime(t)=\left(\gamma_x^\prime(t)\frac{\partial \phi_x}{\partial x}+\gamma_y^\prime(t)\frac{\partial \phi_x}{\partial y}\right) + i\left(\gamma_x^\prime(t)\frac{\partial \phi_y}{\partial x}+\gamma_y^\prime(t)\frac{\partial \phi_y}{\partial y}\right)
\end{aligned}$$
and now, you have to replace with the values above...
