chain rule for complex function

Question

Let $f(t)=x(t)+i y(t),$ where $x,y \colon (a,b) \to \mathbb{R}$ are differentiable on $(a,b).$ Let $F(t)=(f(t))^n$. It can be easily proved, by mathematical induction, that $$\frac{d}{dt}F(t)=n [f(t)]^{n-1}f'(t),$$ provided $n \in \mathbb{N}.$ We regard $F(t)$ as the composition $(g \circ f)(t),$ where $g(z)=z^n$. Then $$F'(t)=\left[\frac{dg}{dz}(f(t))\right]f'(t).$$

Are there any generalized condition on $g$ for this result holds? For example, $g$ is analytic, etc.

Thanks in advance.

Answer 1

Actually, all we need is that $g$ is an holomorphic function. Then we can apply the chain rule and obtain that $F'(t)=g'\bigl(f(t)\bigr).f'(t)$.