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  "body": "\n## Summary\n\nFinite difference coefficients are the weights that turn values of a function on a stencil of points into an approximation of a derivative.\nFor a given derivative and stencil, they are found by matching Taylor expansions.\nThis leads to a Vandermonde system with a unique solution.\nThey are the building blocks of [[numerical differentiation|NVH3F8FTYS]] and of the [[finite difference method|X7HYEBA5SA]] for differential equations.\nTables of central, forward and backward coefficients of various orders are standard.\n\n## Explanations\n\n### L1 human\n\nTo estimate the second derivative of $f$ at $x$ from the values at $x - h$, $x$ and $x + h$, one uses the weights $1, -2, 1$ and divides by $h^2$.\nFor $f(x) = x^2$ at $x = 0$ with $h = 1$, this gives $1 - 0 + 1 = 2$, the true second derivative.\nOther derivatives and other sets of points need other weights.\nFinite difference coefficients are these weights, and there is a systematic way to compute them for any stencil.\n",
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