{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $I \\subseteq \\R$ be an open interval, let $f \\colon I \\to \\R$ be a function, and let $x \\in I$.\nWrite $\\lambda$ for [[Lebesgue measure|2Z2QVWZ262]].\nA measurable set $E \\subseteq \\R$ has *density $1$* at $x$ if\n$$\\lim_{h \\to 0^+} \\frac{\\lambda(E \\cap [x - h, x + h])}{2h} = 1.$$\nThe function $f$ is *approximately differentiable* at $x$, with *approximate derivative* $c \\in \\R$, if the following holds.\nThere is a measurable set $E \\subseteq I$ with $x \\in E$ and density $1$ at $x$ such that\n$$\\lim_{\\substack{y \\to x \\\\ y \\in E \\setminus \\set{x}}} \\frac{f(y) - f(x)}{y - x} = c.$$\nThe approximate derivative is written $f'_{\\mathrm{ap}}(x)$.\n\n## Remarks\n\nExample: let $f(1/n) = 1$ for $n \\ge 2$ and $f(y) = 0$ at all other $y \\in (-1, 1)$.\nAt $x = 0$ the ordinary [[derivative|0F1WR9C83K]] does not exist, since the quotient at $y = 1/n$ equals $n$.\nThe set $E = (-1, 1) \\setminus \\set{1/n : n \\ge 2}$ has density $1$ at $0$, because the removed set is countable, and on $E$ the quotient is $0$; so $f'_{\\mathrm{ap}}(0) = 0$.\n\nNon-example: let $f(y) = 1$ for $y \\ge 0$ and $f(y) = 0$ for $y \u003c 0$, at $x = 0$.\nFor $y \u003c 0$ the quotient is $-1/y$, which tends to $+\\infty$ as $y \\to 0^-$.\nEvery set of density $1$ at $0$ contains points $y \u003c 0$ arbitrarily close to $0$.\nSo $f$ has no approximate derivative at $0$.\n\nIf $f'(x)$ exists, then $f'_{\\mathrm{ap}}(x)$ exists and equals it, by taking $E = I$.\nThe approximate derivative is unique when it exists, because two sets of density $1$ at $x$ meet in a set of density $1$ at $x$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "4W31MKE01V",
    "kind": "definition",
    "msc": [
      "26A24"
    ],
    "name": "Approximate derivative",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Approximate_limit"
      }
    ],
    "tag": "RFK1AVB9H7",
    "uses": [
      "0TXTSEMHR5",
      "0F1WR9C83K"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/RFK1AVB9H7/",
  "links": [
    {
      "field": "defines",
      "target": "4W31MKE01V"
    },
    {
      "field": "uses",
      "target": "0TXTSEMHR5"
    },
    {
      "field": "uses",
      "target": "0F1WR9C83K"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/RFK1AVB9H7.md",
  "msc": [
    "26A24"
  ],
  "name": "Approximate derivative",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RFK/RFK1AVB9H7.md",
  "status": null,
  "tag": "RFK1AVB9H7",
  "uri": "m2://RFK1AVB9H7",
  "url": "https://math2.tamnd.com/wiki/approximate-derivative-definition/",
  "writing": {
    "tag": "RFK1AVB9H7",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}