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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $I \\subseteq \\R$ be an interval, and let $f \\colon I \\to \\R$ be $n$ times continuously differentiable.\nLet $x_0, \\dots, x_n \\in I$ be pairwise distinct.\nLet $\\Sigma_n = \\set{(t_1, \\dots, t_n) \\in \\R^n : t_i \\ge 0, \\ t_1 + \\cdots + t_n \\le 1}$ and set $t_0 = 1 - t_1 - \\cdots - t_n$.\nThen the [[divided difference|N5ZHV4HWDC]] of $f$ is\n$$f[x_0, \\dots, x_n] = \\int_{\\Sigma_n} f^{(n)}(t_0 x_0 + t_1 x_1 + \\cdots + t_n x_n) \\, dt_1 \\cdots dt_n,$$\nwhere the integral is with respect to [[Lebesgue measure|2Z2QVWZ262]] on $\\R^n$.\n\n## Remarks\n\nThe point $t_0 x_0 + \\cdots + t_n x_n$ runs over the convex hull of the nodes, so the integrand is defined.\nFor $n = 1$ the formula reads $\\frac{f(x_1) - f(x_0)}{x_1 - x_0} = \\int_0^1 f'(x_0 + t(x_1 - x_0)) \\, dt$, which is the fundamental theorem of calculus.\nThe general case follows by induction on $n$: integrating over the last variable and using the case $n - 1$ reproduces the recursion that defines divided differences.\n\nSince $\\Sigma_n$ is a [[simplex|YX1DJK2QAG]] of volume $1/n!$, the formula gives $f[x_0, \\dots, x_n] = f^{(n)}(\\xi)/n!$ for some $\\xi$ between the smallest and largest node.\nThis is a form of the [[mean value theorem for divided differences|BQJ9C1CD5X]].\nThe right side is a continuous function of $(x_0, \\dots, x_n)$ even when nodes coincide.\nSo it serves as the definition of divided differences with repeated nodes, which is what [[Hermite interpolation|9Z50M2WPTP]] uses.\n\n## Explanations\n\n### L1 human\n\nA divided difference is a ratio of differences of values.\nThe formula says that it is also an average of a high derivative of $f$, taken over all weighted mixtures of the nodes.\nSo divided differences are smoothed derivatives.\n\n### L2 human\n\nThe formula shows at once that $f[x_0, \\dots, x_n]$ is symmetric in the nodes and depends continuously on them.\nLetting all nodes tend to one point $x$ gives $f[x, \\dots, x] = f^{(n)}(x)/n!$, the Taylor coefficient.\nIt is the bridge between [[Newton interpolation|9W77FPTM9E]] and [[Taylor polynomials|0HFSFVMWQB]].\n",
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