central-difference formula

Approximating a derivative from one point ahead and one point behind, straddling x symmetrically -- one order of accuracy better than using either side alone.

The central-difference formula approximates $f'(x)$ by $D(h)=\dfrac{f(x+h)-f(x-h)}{2h}$ , the slope of the secant line through $(x-h,f(x-h))$ and $(x+h,f(x+h))$ . Its truncation error is $O(h^2)$ -- one order better than the one-sided forward- or backward-difference formulas -- because the symmetric expansion cancels the $h^1$ error term exactly (ch.05 sec.3).

Examples

For $f(x)=x^2$ at $x=1$ , $h=0.1$ : $\dfrac{f(1.1)-f(0.9)}{0.2}=\dfrac{1.21-0.81}{0.2}=2$ , exactly matching $f'(1)=2$ -- no error at all, since $x^2$ 's own error term vanishes identically.