For numerical differentiation of $\frac{df}{dx}$ at $x=0$, suppose I use finite difference methods of $p^{th}$ order, which produces $O(h^p)$ error, where $h$ is the step size I use for numerical differentiation. From what I see, I can find error bounds (no, $O (h^{p})$ is just asymptotic error bound, and what I want is more specific) for 1st-order, 2nd-order, 3rd-order finite difference, but not for general $p^{th}$ order case. What would be the formula for the $p^{th}$-order error bound?
Also, would there be a better numerical differentiation than finite difference methods in terms of error bound and what would be error bounds? (for example, $f(x) = \cos(x)$)
I will accept any answer that answers the first question - I am fine having an answer to the first question only.