  The methods commonly used for numerical differentiation, such as the
"center-difference formula" and "four-points formula" are unusable in
simulations or real-time data analysis because they require knowledge of the
future. In Bard'11, an algorithm was shown that generates formulas that require
knowledge only of the past and present values of $f(t)$ to estimate $f'(t)$.
Furthermore, the algorithm can handle irregularly spaced data and higher-order
derivatives. That work did not include a rigorous proof of correctness nor the
error bounds. In this paper, the correctness and error bounds of that algorithm
are proven, explicit forms are given for the coefficients, and several
interesting corollaries are proven.
