{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-derivative-rules",
  "semantic_family_id": "sf-derivative-rules",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "多項式的和、乘積與合成微分"
  },
  "summary": {
    "zh-Hant": "四條微分規則先保留可微前提；本輪只在有限整數係數多項式上精確計算。"
  },
  "statement": {
    "zh-Hant": "在相應點可微的實函數有 (f+g)'=f'+g'、(cf)'=cf'、(fg)'=f'g+fg'；若g在x可微且f在g(x)可微，(f∘g)'(x)=f'(g(x))g'(x)。本輪f、g是全R多項式，以常數項起算的升冪係數列表表示。"
  },
  "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": "題庫輸入多項式每列1至4個安全整數係數，即次數至多3；末端零係數正規化，空列不合法。"
    },
    {
      "id": "a2",
      "zh-Hant": "所有輸入多項式在R處處可微，合成的外函數在g(x)取導數，不在x錯置；不以此切片判任意函數可微。"
    },
    {
      "id": "a3",
      "zh-Hant": "主端展開和、縮放、乘積或合成多項式再求導；出題端用規則另算。合成／乘積可能升次，不能截掉新項。"
    }
  ],
  "symbols": [
    {
      "token": "D",
      "role": "polynomial_derivative_at_point",
      "role_zh": "指定多項式導數的點值",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "real_polynomial_function",
      "role_zh": "合併後或指定的多項式函數",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "real_evaluation_point",
      "role_zh": "整數題庫中的實數求值點",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-derivative",
      "title_zh": "實函數在一點的導數與差商",
      "reason_zh": "先保留導數存在性語義。"
    },
    {
      "id": "mko-polynomial",
      "title_zh": "多項式",
      "reason_zh": "升冪係數及有限運算。"
    },
    {
      "id": "mko-composition",
      "title_zh": "函數合成",
      "reason_zh": "外函數輸入為g(x)。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "乘積規則不是f'g'；鏈式規則既要外導數在內函數值取值，也要乘內導數。共同常數項只在確實不影響指定規則時才可去除。",
      "升冪列表 [a0,a1,a2] 表示a0+a1*x+a2*x²，不是降冪。末端補零只是同一個多項式，不能當新題。",
      "零純量、零多項式、常數外函數或常數內函數需要保留；合成後是常數時導數零。不同答案的真實尺度與求值點不會因去重被盲目刪除。"
    ],
    "common_misconception_zh": "乘積導數直接乘兩導數、鏈式規則漏內導數或外函數取錯點，或把係數序列反讀。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限多項式恆等與點值可精確重算；一般微分規則的形式 Evidence仍未建立。",
    "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": "固定版 polynomial derivative/eval、HasDerivAt.add／mul／comp橋接；不以有限點值替代一般可微前提。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-derivative-rules.md 整理；2026-10-07 脈衝 R05（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"
  }
}