{
  "aliases": [
    "sum-rule-for-derivatives"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0MH1WK89PP"
    },
    {
      "field": "uses",
      "target": "0Z9JB71BS0"
    }
  ],
  "body": "\n## Statement\n\nLet $f$ and $g$ be real functions on an open interval $I$, both differentiable at $a \\in I$.\nLet $\\alpha$ and $\\beta$ be real numbers.\nThen $\\alpha f + \\beta g$ is differentiable at $a$, and\n$$(\\alpha f + \\beta g)'(a) = \\alpha f'(a) + \\beta g'(a).$$\n\n## Remarks\n\nThe difference quotient of $\\alpha f + \\beta g$ is $\\alpha$ times that of $f$ plus $\\beta$ times that of $g$. The limit laws give the result.\nThe case $\\alpha = \\beta = 1$ is the sum rule $(f + g)' = f' + g'$. The case $\\beta = 0$ is the [[constant multiple rule|0Z9JB71BS0]].\nMathlib states the sum rule as `deriv_add`.\n\n## Explanations\n\n### L1 human\n\nDifferentiation respects sums and constant factors.\nSo we can differentiate a sum term by term.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "sum-rule-for-derivatives"
    ],
    "kind": "statement",
    "msc": [
      "26A24"
    ],
    "name": "Linearity of differentiation",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q4895685",
      "wikipedia": "Linearity of differentiation"
    },
    "sources": [
      {
        "locator": "Statement and derivation",
        "url": "https://en.wikipedia.org/wiki/Linearity_of_differentiation"
      }
    ],
    "tag": "0M0TKFRX2A",
    "uses": [
      "0F1WR9C83K",
      "0RAM4E07NR"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0M0TKFRX2A/",
  "links": [
    {
      "field": "uses",
      "target": "0F1WR9C83K"
    },
    {
      "field": "uses",
      "target": "0RAM4E07NR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0M0TKFRX2A.md",
  "msc": [
    "26A24"
  ],
  "name": "Linearity of differentiation",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0M0/0M0TKFRX2A.md",
  "status": null,
  "tag": "0M0TKFRX2A",
  "uri": "m2://0M0TKFRX2A",
  "url": "https://math2.tamnd.com/wiki/linearity-of-differentiation/",
  "writing": {
    "tag": "0M0TKFRX2A",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}