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  "body": "\n## Summary\n\nThe symmetric derivative of a function at a point is a limit of slopes of secant lines.\nThe two points of each secant line lie symmetrically on both sides of the point.\nWhen the ordinary [[derivative|0F1WR9C83K]] exists, the symmetric derivative exists and has the same value.\nThe converse fails, so the symmetric derivative is a genuine generalization.\nIts quotient is also the central difference used to approximate derivatives numerically.\n\n## Explanations\n\n### L1 human\n\nInstead of comparing $f(x + h)$ with $f(x)$, compare $f(x + h)$ with $f(x - h)$, and divide by the distance $2h$.\nThis balances the left and the right side.\nFor $\\abs{x}$ at $0$, the two sides cancel: $\\abs{h} - \\abs{-h} = 0$.\nSo $\\abs{x}$ has symmetric derivative $0$ at $0$, although it has a corner there.\n",
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