derivative of complex composite functions I want to derivate a real-valued function of real variable, defined as:
$$L(x) = f(g(x)), $$
where $g$ is a complex-valued function of real variable and $f$ a real-valued function of complex variable. 
The classical way suggests $L'(x)=g'(x)f'(g(x))$, but what is the meaning of $f'$? And $g'$ is complex so the derivative of $L$ is complex (what seems absurd)?
Thanks for your answers
 A: There is actually no good notion of $f'(z)$, which is a consequence of complex differentiability. If $f=u+iv$ were complex differentiable, we would require that $u_x=v_y$ and $u_y=-v_x$, which are the Cauchy Riemann Equations. However, we have $v=0$, since $f$ is entirely real, so $u_x=u_y=0$. This can only happen if $u$ is a constant function, so if $f$ is anything other than a real constant, then $f'$ has little to no meaning.
One way of looking at this, however, is to not think of $f$ and $g$ as having anything to do with complex numbers, but instead thinking of $g$ as mapping into $\mathbb{R}^2$ and $f$ as a function on a subset of $\mathbb{R}^2$. We then define
$$g(x)=u(x)+iv(x)\implies g(x)=(u(x),v(x))$$
From here, we have
$$f(z)=f(u+iv)\implies f(z)=F(u,v)$$
where we have implicitly defined a new function $F$ as a function that operates on the real and imaginary parts of $z$ as two separate variables. Putting this all together, we have
$$L(x)=F(u(x),v(x))$$
and so,
$$L'(x)=\frac{\partial F}{\partial u}u'(x)+\frac{\partial F}{\partial v}v'(x)$$
where
$$f(z)=F(\Re(z),\Im(z)),\;\;\;\;g(x)=u(x)+iv(x)$$
A: why should L' be komplex? since f ist real f' is real so f'*g' should be real? and the Rule L'=f'(g)*g' applies.
A: $f'$ has no meaning in this context since a complex differentiable function cannot have just the real numbers as it's range. But we can think of f as being a function of two variables, $f(z,\bar{z})$ and so we can fix our chain rule:
$$L'(x) = \partial_z f(g(x),\bar{g}(x)) \cdot g'(x) + \partial_{\bar{z}} f(g(x),\bar{g}(x)) \cdot \bar{g}'(x)$$
Where the usual defintions of the partials are given by $\partial_z = \frac{1}{2}(\partial_x - i\partial_y)$ and $\partial_{\bar{z}} = \frac{1}{2}(\partial_x +i\partial_y)$, where we think of a function of a complex variable as being functions of real variables ($z=x+iy$). 
Under this defintion everything is sensible and will come out to a real number.
