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  "body": "\n## Summary\n\nThe approximate derivative of a real function at a point is a limit of its difference quotients.\nIn this limit one may ignore a set that is very thin near the point.\nThe ignored set must have density $0$ at the point, in the sense of [[Lebesgue's density theorem|WSVXYSNHNX]].\nApproximate derivatives exist in many cases where ordinary derivatives do not, and they are the derivatives used in the [[Khinchin integral|Q1TV4F608D]].\n\n## Explanations\n\n### L1 human\n\nAn ordinary derivative looks at all points near $x$.\nAn approximate derivative is allowed to skip a set of points, as long as that set takes up a smaller and smaller share of each small interval around $x$.\nFor example, change the zero function to the value $1$ at the points $1, \\frac12, \\frac13, \\dots$.\nThe new function has no derivative at $0$, but its approximate derivative at $0$ is $0$, since the changed points form a countable set, which is invisible to measure.\n",
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