Q:

Derivative of the conjugate of a function

In short, is the derivative of the conjugate of a function, the conjugate of the derivative of that function?
We assume that the function is complex-valued.
I think this property is true when the derivation variable is real, but my question is: if this is the case, can't we always do a change of variable so that we change from a real variable to a complex one? It's a bit confusing to me.

A:

In all non-trivial cases the conjugate of a complex-differentiable function is not complex-differentiable  itself, so it makes no sense talking about its derivative. To see this, let $f(z)$ be a non-constant complex-differentiable function. Then $f(z)$ and $\overline{f(z)}$ agree on this uncountable set with a limit point:
$$\{z\in\Bbb C\mid \Im (f(z))=0\}.$$
For complex differentiable functions, this suffices to show that they are equal which is a contradiction because they are non-constant, hence cannot be real-valued. (I honestly forgot the name of the theorem stating the equality).

But if you only talk about complex valued functions $f:\Bbb R\to\Bbb C$, then yes, $\overline{f'(x)}=(\overline{f(x)})'$. You can see this via linearity. Differentiation commutes with $i$. Let $f(x)=a(x)+i b(x)$, then
$$(\overline{f(x)})'=(a(x)-i b(x))'=a'(x)-(i b(x))'=a'(x)-ib'(x)=\overline{f'(x)}.$$

