I'm interested in approximating higher derivatives of a function via values of the function only. I guess the following question has been studied, but I haven't been able to find a reference. I know that one can, via repeated application of the finite difference formula $$f'(x)\approx \frac{f(x+\delta)-f(x)}{\delta}$$ approximate the higher derivatives using equally spaced nodes as $$f^{(k)}(x)\approx \frac{1}{\delta^k}\left(\sum_{i=0}^k(-1)^i{k\choose i}f(x+(k-i)\delta)\right)$$

and that under appropriate assumptions on $f$ this can be made provably close to the actual $f^{(k)}(x)$ in a quantitative way (depending on $\delta$) (of course, without assumptions, the derivative can be made to be anything we want by Hermite interpolation).

But is there a place discussing the need for $\Omega(k)$ calls to $f$, or providing some sort of lower bound?