I don't understand what the fuzz is about all this forward-mode AD stuff. This is the easy part. Hell, you can basically use complex numbers to do all this for you for free (use Cauchy-rule + FD) without changing the rest of your code.
tldr: f‘(x0)≈Im(f(x0+hi))/h
Choice of h dictates precision (less bias than doing it purely in the real domain, but still .. a nontrivial choice). And also requires your functions to be analytical - which many aren't.
Whereas autodiff is precise .