I am studying Newton's method modified by the book Zorich, Mathematical analysis II, page 39,40:
It seems to me, if I make no mistakes, that there is a problem in the derivative of $ A (x) $. The author says that $ | A '(x) | = | [f' (x_0)] ^ {- 1} \cdot f '(x) | $, while I would say that:
$$ | A '(x) | = | 1- [f' (x_0)] ^ {- 1} \cdot f '(x) | $$
Am I wrong?
Yes, your observation is correct. This is also confirmed by the fact that this corrected expression is the smaller the closer $f'(x_0)$ is to $f'(x)$, that is, the closer the step is to the Newton method.
Perhaps they mixed this up with the derivative of the Newton step where this first term indeed cancels, $$ N(x)=x-[f'(x)]^{-1}f(x)\implies N'(x)=I-I+[f'(x)]^{-1}[f''(x)][f'(x)]^{-1}f(x) $$