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  "body": "\n## Summary\n\nDivided differences are quotients built recursively from the values of a function at a list of points.\nFirst comes the slope between two points, then the difference of two slopes divided by the spread of three points, and so on.\nThe divided difference $f[x_0, \\dots, x_n]$ is the leading coefficient of the polynomial of degree at most $n$ that interpolates $f$ at the points.\nThey are the coefficients of the [[Newton form|9W77FPTM9E]] of the interpolation polynomial.\nFor smooth functions they behave like scaled derivatives: by the [[mean value theorem for divided differences|BQJ9C1CD5X]], $f[x_0, \\dots, x_n] = f^{(n)}(\\xi)/n!$ for some $\\xi$.\n\n## Explanations\n\n### L1 human\n\nThe first divided difference $\\frac{f(x_1) - f(x_0)}{x_1 - x_0}$ is the slope of the chord between two points of a graph.\nThe second one compares two neighbouring slopes and so measures how the graph bends.\nFor $f(x) = x^2$ and the points $0, 1, 3$, the slopes are $1$ and $4$, and the second divided difference is $\\frac{4 - 1}{3 - 0} = 1$, the leading coefficient of $x^2$.\nHigher divided differences continue the same pattern.\nThe [[definition|T942SHWSSF]] gives the recursion.\n",
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