Suppose I have $n$ *samples* $(x_i, f(x_i))_{i=1}^n$ from an *unknown* function $f$. **I need to approximate (estimate) the derivative $f'(x^*)$ at some new test point $x^*$**, that is not necessarily one of the $x_i$. I am assuming nothing about the $x_i$: They can be regularly or irregularly sampled, have large gaps, etc.

Naively, numeric differentiation seems like the *only* option. The "other" option would be automatic differentiation, but from my understanding of this you have to actually know $f$ in order to use auto-differentiation.

*Suprisingly, in all of the auto-diff tutorials I have come across, this assumption is never mentioned.*

So, now I wonder if there is some way to apply auto-diff to my problem that I haven't come up with. (Or, some other method altogether besides numerical differentiation.) General references on this problem are welcome as well!