Central difference

Summary

The central difference of a function $f$ with step $h$ is $\delta_h f(x) = f(x + \tfrac{h}{2}) - f(x - \tfrac{h}{2})$. It is symmetric around the point $x$. For a smooth function, the quotient $\delta_h f(x) / h$ approximates $f'(x)$ with an error of order $h^2$, better than the order $h$ of the one-sided quotients.