Suppose we analyze the order of accuracy of a finite difference approximation of a derivative,
$$f'(x)=\frac{1}{2h} \left[f(x-2h) -4f(x-h) +3f(x)\right]$$
and we conclude that the order of accuracy is 1 because our analysis would indicate that there is one $O(h)$ term.
Now, what if instead of the above formula we tried to analyze
$$f'(x)=\frac{1}{2h} \left[f(x-2h) -4f(x-h) +4f(x)\right]$$
Then our analysis would give us an additional (to $O(h)$) term, $\frac{f(x)}{2h}$, or $O(h^{-1})$.
What would this signify in terms of the order of accuracy?