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  "body": "\n## Definition\n\nLet $k \\ge 1$ be an integer, let $f$ be a real function defined near a point $x \\in \\R$, and let $s_1, \\ldots, s_N$ be distinct real numbers.\nA *numerical differentiation formula* for the [[$k$-th derivative|04TKTF8KDK]] $f^{(k)}(x)$ is an expression\n$$D_h f(x) = \\frac{1}{h^k} \\sum_{j=1}^{N} a_j \\, f(x + s_j h), \\qquad h \u003e 0,$$\nwith real coefficients $a_1, \\ldots, a_N$ that do not depend on $h$ or $f$.\nIt has *order* $p \\ge 1$ if $D_h f(x) - f^{(k)}(x) = O(h^p)$ as $h \\to 0$ for every $f$ of class $C^{k+p}$ near $x$, in [[big O notation|0TGV80E6A4]].\n*Numerical differentiation* is the approximation of $f^{(k)}(x)$ by such formulas, or more generally by derivatives of functions fitted to finitely many values of $f$.\n\n## Remarks\n\nExample: the central difference $D_h f(x) = \\frac{f(x + h) - f(x - h)}{2h}$ for $k = 1$, with stencil $-1, 1$ and $a = (-\\tfrac12, \\tfrac12)$, has order $2$.\nCheck with Taylor's theorem for $f \\in C^3$: $f(x \\pm h) = f(x) \\pm h f'(x) + \\frac{h^2}{2} f''(x) \\pm \\frac{h^3}{6} f'''(\\xi_\\pm)$, so $D_h f(x) - f'(x) = O(h^2)$.\nThe forward difference $\\frac{f(x + h) - f(x)}{h}$ has only order $1$, with error $\\frac{h}{2} f''(\\xi)$.\nNon-example: $\\frac{f(x+h) + f(x)}{h}$ approximates no derivative; for $f = 1$ it equals $\\frac{2}{h}$, which does not tend to $f'(x) = 0$.\nIn floating-point arithmetic with unit round-off $u$, the forward difference has total error about $\\frac{h}{2}\\abs{f''} + \\frac{2u\\abs{f}}{h}$, which is smallest for $h$ of the order of $\\sqrt{u}$.\n\n## Explanations\n\n### L1 human\n\nA numerical differentiation formula combines a few nearby values of a function to estimate one of its derivatives.\nThe order says how fast the error shrinks when the spacing $h$ shrinks: order $2$ means that halving $h$ divides the error by about $4$.\n",
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