{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "01NN8NY497"
    },
    {
      "field": "uses",
      "target": "01V9FPBBVK"
    },
    {
      "field": "uses",
      "target": "05RCT1522R"
    },
    {
      "field": "uses",
      "target": "0EY05Z9RPN"
    },
    {
      "field": "uses",
      "target": "0J8T4K4YTE"
    },
    {
      "field": "uses",
      "target": "0NNQY8WCW6"
    },
    {
      "field": "uses",
      "target": "0PJXBR6QA3"
    },
    {
      "field": "uses",
      "target": "0SCSQWXRA4"
    },
    {
      "field": "uses",
      "target": "0TCBZMEYGN"
    }
  ],
  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f \\colon U \\to \\mathbb{R}$, and let $i, j \\in \\{1, \\dots, n\\}$.\nSuppose that the [[partial derivative|05AXG3QRKR]] $\\partial f / \\partial x_i$ exists at each point of $U$.\nThe second partial derivative of $f$ with respect to $x_i$ and then $x_j$ at $a \\in U$ is\n$$\\frac{\\partial^2 f}{\\partial x_j \\, \\partial x_i}(a) = \\frac{\\partial}{\\partial x_j}\\left(\\frac{\\partial f}{\\partial x_i}\\right)(a),$$\nwhen this partial derivative exists. It is also written $f_{x_i x_j}(a)$ or $\\partial_j \\partial_i f(a)$.\nFor $i = j$, one writes $\\partial^2 f / \\partial x_i^2$.\nBy induction, a partial derivative of order $k \\ge 2$ is a partial derivative of a partial derivative of order $k - 1$: $\\dfrac{\\partial^k f}{\\partial x_{i_k} \\cdots \\partial x_{i_1}} = \\dfrac{\\partial}{\\partial x_{i_k}} \\left( \\dfrac{\\partial^{k-1} f}{\\partial x_{i_{k-1}} \\cdots \\partial x_{i_1}} \\right)$.\n\n## Remarks\n\nIn $\\partial^2 f / \\partial x_j \\, \\partial x_i$ the variable nearest to $f$ is differentiated first. In $f_{x_i x_j}$ the variable written first is differentiated first.\nFor $f(x, y) = x^3 y^2$, one gets $f_{xx} = 6 x y^2$, $f_{xy} = f_{yx} = 6 x^2 y$ and $f_{yy} = 2 x^3$.\nIf the second partial derivatives are continuous, then the order of differentiation does not matter, by [[Schwarz's theorem|0NNQY8WCW6]]. Without continuity, the mixed partial derivatives [[can differ|01NN8NY497]].\nThe second partial derivatives at a point form the [[Hessian matrix|0693Y2METT]].\n\n## Explanations\n\n### L1 human\n\nA second partial derivative measures how a slope changes. For example, $f_{xy}$ measures how the slope in the $x$-direction changes as you move in the $y$-direction.\nYou compute it by differentiating twice, one variable at a time.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "09ZXGX55DM",
    "kind": "definition",
    "msc": [
      "26B05"
    ],
    "name": "Higher partial derivatives",
    "sources": [
      {
        "locator": "Higher order partial derivatives",
        "url": "https://en.wikipedia.org/wiki/Partial_derivative"
      }
    ],
    "tag": "02RSXVM1ZN",
    "uses": [
      "05AXG3QRKR"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/02RSXVM1ZN/",
  "links": [
    {
      "field": "defines",
      "target": "09ZXGX55DM"
    },
    {
      "field": "uses",
      "target": "05AXG3QRKR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/02RSXVM1ZN.md",
  "msc": [
    "26B05"
  ],
  "name": "Higher partial derivatives",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/02R/02RSXVM1ZN.md",
  "status": null,
  "tag": "02RSXVM1ZN",
  "uri": "m2://02RSXVM1ZN",
  "url": "https://math2.tamnd.com/wiki/higher-partial-derivatives-definition/",
  "writing": {
    "tag": "02RSXVM1ZN",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}