Solution
= Solution
Apply the preceding <Taylor's theorem> at $t=h$ and $t=-h$, with $h>0$. The even-order terms cancel in the <central finite difference>, giving
$$
d(x,h)-f'(x)=\frac{R_x(h)-R_x(-h)}{2h}.
$$
Both <Lagrange remainders> have magnitude at most $h^3/6$, so the <triangle inequality> gives the <sharp central-secant derivative error>
$$
\boxed{|d(x,h)-f'(x)|\le\frac{h^2}{6}.}
$$