{
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    "name": "Manazoo",
    "url": "https://manazoo.com/"
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  "license": "All rights reserved（AI の参照・要約と出典リンクつきの引用は可。転載・再配布・販売は不可）",
  "terms_url": "https://manazoo.com/terms/",
  "disclaimer": "内容の正確さには気をつけていますが、すべてが正しいとは限りません。テストや受験の前には、教科書や先生の説明でも確かめてください。間違いを見つけたら意見箱から知らせてください。",
  "attribution": "事実データの一部は Wikidata（CC0）による。単元の構成は文部科学省「中学校学習指導要領」「高等学校学習指導要領」（政府標準利用規約2.0）。",
  "kind": "entity",
  "slug": "differential-coefficient-and-derivative",
  "name": "微分係数と導関数",
  "entity_kind": "concept",
  "wikidata_qid": "Q29175",
  "wikidata_url": "https://www.wikidata.org/wiki/Q29175",
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  "course": {
    "slug": "math-2",
    "title": "数学Ⅱ",
    "url": "https://manazoo.com/high/math-2/"
  },
  "unit": {
    "slug": "differential-and-integral-calculus",
    "title": "微分・積分の考え",
    "url": "https://manazoo.com/high/math-2/differential-and-integral-calculus/"
  },
  "period": null,
  "place": null,
  "summary_points": [
    "平均変化率(f(a+h)−f(a))/hのhを0に近づけた極限値が、x＝aでの微分係数f′(a)。",
    "aをxに置きかえた式f′(x)を導関数といい、xⁿの導関数はnxⁿ⁻¹（n≦3が中心）。",
    "定数倍・和・差の導関数は、それぞれの関数の導関数を定数倍・和・差すればよい。"
  ],
  "facts": null,
  "key_points": [
    {
      "term": "平均変化率",
      "note": "xがaからa＋hまで変わるときの、yの変化の割合"
    },
    {
      "term": "微分係数",
      "note": "平均変化率のhを0に近づけた極限値f′(a)"
    },
    {
      "term": "導関数",
      "note": "微分係数を一般のxで表した式f′(x)"
    }
  ],
  "summary": null,
  "causality": null,
  "timeline": [],
  "related": {
    "cause": [],
    "result": [],
    "people": [],
    "same_period_japan": []
  },
  "questions": [
    {
      "id": "q-dcd-average-rate-x2",
      "type": "input",
      "prompt": "f(x)＝x² について、x＝1からx＝3までの平均変化率を求めよ。",
      "choices": null,
      "answer": "4",
      "accept": [],
      "explanation": "平均変化率は(f(3)−f(1))/(3−1)で求める。f(3)＝9、f(1)＝1なので、(9−1)/2＝4。",
      "entities": [
        "differential-coefficient-and-derivative"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-dcd-derivative-value-at-1",
      "type": "input",
      "prompt": "f(x)＝2x³−3x²＋1 の導関数f′(x)を使って、f′(1)の値を求めよ。",
      "choices": null,
      "answer": "0",
      "accept": [],
      "explanation": "f′(x)＝6x²−6xなので、f′(1)＝6×1−6×1＝0。",
      "entities": [
        "differential-coefficient-and-derivative"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-dcd-limit-choice",
      "type": "choice",
      "prompt": "f(x)＝x² のx＝2での平均変化率(f(2+h)−f(2))/hは、hを0に近づけるとどの値に近づくか。",
      "choices": [
        "0",
        "2",
        "4",
        "8"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "(x²)′＝2xなので、x＝2での微分係数はf′(2)＝2×2＝4。平均変化率(2＋h)²−2²をhで割ると4＋hになり、hを0に近づけると4に近づく。",
      "entities": [
        "differential-coefficient-and-derivative"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-dcd-power-rule-choice",
      "type": "choice",
      "prompt": "導関数の公式(xⁿ)′＝nxⁿ⁻¹を使うと、(x³)′はどれになるか。",
      "choices": [
        "3x",
        "3x²",
        "x³",
        "x²"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "n＝3を(xⁿ)′＝nxⁿ⁻¹に当てはめると、(x³)′＝3x³⁻¹＝3x²になる。",
      "entities": [
        "differential-coefficient-and-derivative"
      ],
      "verified": true,
      "placeholder": false
    }
  ],
  "sources": [
    {
      "label": "文部科学省「高等学校学習指導要領（平成30年告示）解説 数学編 理数編」（数学Ⅱ (5)微分・積分の考え ア(ｱ) 微分係数や導関数の意味について理解し、関数の定数倍、和及び差の導関数を求めること・［内容の取扱い］(1) 微分係数や導関数を求める際に必要となる極限は直観的に理解させる。解説 p.64〜65）",
      "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf"
    }
  ],
  "media": [
    {
      "src": "/media/differential-coefficient-and-derivative/secant-to-tangent.png",
      "alt": "y＝x²のグラフで、x＝1からhだけ進んだ点を結ぶ割線がhを0に近づけると接線（傾き2）に近づくようすを3本の割線で示した図",
      "license": "オリジナル（コードで作図）",
      "author": "Asatte Studio（Manazoo）",
      "source_url": "https://manazoo.com/credits/",
      "section": "formula",
      "caption": "割線の傾き（平均変化率）はh→0で微分係数2に近づく"
    },
    {
      "src": "/media/differential-coefficient-and-derivative/falling-object-velocity.png",
      "alt": "s＝4.9t²のグラフと、t＝1からt＝1＋hまでの平均の速さをh＝1・0.1・0.01で計算した値が9.8m/秒に近づくようすを示した図",
      "license": "オリジナル（コードで作図）",
      "author": "Asatte Studio（Manazoo）",
      "source_url": "https://manazoo.com/credits/",
      "section": "why",
      "caption": "s＝4.9t²のt＝1での瞬間の速さは9.8m/秒"
    }
  ],
  "status": "published",
  "placeholder": false,
  "updated_at": "2026-10-22"
}