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  "body": "\n## Summary\n\nThe Grünwald–Letnikov derivative of real order $\\alpha$ is a limit of weighted sums of values of the function.\nIt extends the [[difference quotient|AZBDB5F4M4]] that defines ordinary derivatives.\nThe weights are generalised [[binomial coefficients|0GBZM0MT0F]] $\\binom{\\alpha}{k}$ with alternating signs.\nFor whole orders it gives the usual derivatives, and for smooth enough functions it agrees with the Riemann–Liouville derivative of [[fractional calculus|T73XK15SA2]].\nIts discrete form is the basis of many numerical methods for fractional differential equations.\n\n## Explanations\n\n### L1 human\n\nThe first derivative is the limit of $(f(x) - f(x - h))/h$, and the second derivative is the limit of $(f(x) - 2f(x - h) + f(x - 2h))/h^2$.\nThe coefficients $1, -1$ and $1, -2, 1$ are binomial coefficients with alternating signs.\nBinomial coefficients $\\binom{\\alpha}{k}$ make sense for any real $\\alpha$, so the same recipe defines a derivative of order $\\alpha$.\nFor non-integer $\\alpha$ the sum does not stop, so the derivative at $x$ depends on all earlier values of $f$ back to a starting point.\n",
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