= Step-size choice for a noisy central derivative
{title2=$h_*=(3\delta/M)^{1/3}$}
Suppose $|f'''|\le M$ and each sampled value of $f$ has <absolute error> at most $\delta$. For a <central finite difference> with $h>0$, the <triangle inequality> and the <sharp central-secant derivative error> give
$$
|\widetilde d-f'|\le\frac{Mh^2}{6}+\frac\delta h.
$$
For $M,\delta>0$, the <derivative> of this bound is $Mh/3-\delta/h^2$. It changes sign at $h_*=(3\delta/M)^{1/3}$, proving the unique minimum, of value $(3\delta)^{2/3}M^{1/3}/2$. This balance explains why reducing the step indefinitely amplifies sample <absolute error> even as <truncation error> decreases.
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