From Munkres:
Any idea why my book writes $f’(x;v)$? The derivative is a function from $A\to \mathbb R$ after all, so how can it suddenly have an argument of $(x;v)$? It looks like they're using some sort of chain rule, but I don't really see it. Any clarification on $f'(x;v)$ is appreciated.
The derivative, at a point $x$, is a linear map $L:\mathbb{R}^n \to \mathbb{R}$.
This linear map depends, of course, on the point $x$. So it is common to see notations like $L_x$, or $f'_x$ etc.
Munkres seems to be putting that dependence in an argument, which is essentially just a stylistical choice (when talking about derivatives). Formally, he is considering the map \begin{align*} f': A \times \mathbb{R}^n &\to \mathbb{R} \\ (x,v) &\mapsto L_x\cdot v. \end{align*}