{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-derivative",
  "semantic_family_id": "sf-derivative",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實函數在一點的導數與差商"
  },
  "summary": {
    "zh-Hant": "導數是非零步差商的兩側極限，不是任何一個有限步差商；絕對值尖點另保留不存在。"
  },
  "statement": {
    "zh-Hant": "對f:R→R，f在x的導數是lim_(h→0,h≠0)(f(x+h)−f(x))/h，若此兩側有限極限存在。本輪仿射及二次以下多項式的精確導數，以及 |t−c| 在t=c左右導數不同、導數不存在的邊界。"
  },
  "formula": {
    "tex": "D(f,x)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>D</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "D",
        "glyph": "D"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "f",
          "glyph": "f"
        },
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "d6ea9099addaa5ab668567e8f1a09463ba78acd6d799c710dbc3c964a40637a3"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "全實數論域、中心點值及兩側鄰域均合法；有限步h須非零。"
    },
    {
      "id": "a2",
      "zh-Hant": "f(t)=a2*t²+a1*t+a0允許a2=0，指二次以下多項式，不能在這時仍宣稱二次函數的非零係數資格。"
    },
    {
      "id": "a3",
      "zh-Hant": "此多項式導數為2a2*x+a1；有限差商為該值+a2*h。|t−c|在t>c為1、t<c為−1，t=c不存在，不填0。"
    }
  ],
  "symbols": [
    {
      "token": "D",
      "role": "real_derivative_at_point",
      "role_zh": "實函數在指定點的導數",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "real_total_function",
      "role_zh": "指定實數總函數",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "real_input",
      "role_zh": "導數中心點",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-function-limit",
      "title_zh": "去心實數論域的函數極限",
      "reason_zh": "非零步差商的兩側極限。"
    },
    {
      "id": "mko-rate-of-change",
      "title_zh": "兩點間的平均變化率",
      "reason_zh": "有限差商不是瞬時導數。"
    },
    {
      "id": "mko-polynomial",
      "title_zh": "多項式",
      "reason_zh": "二次以下係數模型。"
    },
    {
      "id": "mko-absolute-value",
      "title_zh": "絕對值",
      "reason_zh": "尖點的兩側仿射分段。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "f(x)是中心值，h=0時不能直接把分母清零求差商。先處理h≠0的代數式，再看其極限是否存在。",
      "對二次以下式，有限差商與導數可相差a2*h；同一模型的特殊有限步剛好相等，不把一般有限差商都升格為導數。常數與仿射是合法低階邊界。",
      "在絕對值中心，h>0與h<0的差商分別為1與−1，所以沒有兩側導數。形式庫可能把deriv設為全定義並在無導數時填預設值；本條目的存在性判定不能因此填0。"
    ],
    "common_misconception_zh": "有限差商直接叫導數、步h=0照算，或在尖點以全定義API的預設0冒充導數。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "已宣告模型的精確代數與尖點判定可核對；一般微分及形式導數 Evidence未新增。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "綁定 HasDerivAt及nonzero slope limit，保留存在性與totalized deriv的差異。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-derivative.md 整理；2026-10-07 脈衝 R04（使用者手動接續 2026-10-08 工作序列；實際日期仍為 2026-10-07），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-07"
  }
}