{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-partial-derivative",
  "semantic_family_id": "sf-partial-derivative",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "偏導數"
  },
  "summary": {
    "zh-Hant": "偏導數固定其他座標，只沿一個座標切片取h趨近零的單變數導數。"
  },
  "statement": {
    "zh-Hant": "二變數函數f在點(x,y)的x偏導，以y固定並令h≠0趨近0的差商極限定義；y偏導類似。本輪對二次多項式做精確符號微分，不用單一有限差分代替極限。"
  },
  "formula": {
    "tex": "D_i(f)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi>D</mi><mi>i</mi></msub><mo>(</mo><mi>f</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "subscript",
        "base": {
          "type": "symbol",
          "name": "D",
          "glyph": "D"
        },
        "subscript": {
          "type": "symbol",
          "name": "i",
          "glyph": "i"
        }
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "f",
          "glyph": "f"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "ee062954113fc441911b2bb598a5698218ff7fade5c66174ac2fb92f9938e86f"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "係數固定依[1,x,y,x²,xy,y²]排序，題庫使用整數係數與整數點。"
    },
    {
      "id": "a2",
      "zh-Hant": "多項式切片全域有定義；偏導存在仍不被用來推出一般函數連續或全微分。"
    }
  ],
  "symbols": [
    {
      "token": "D_i",
      "role": "partial_derivative",
      "role_zh": "座標偏導",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "multivariable_real_function",
      "role_zh": "二變數實函數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-multivariable-real-function",
      "title_zh": "多變數實函數",
      "reason_zh": "偏導建立在明示座標與定義域上。"
    },
    {
      "id": "mko-derivative",
      "title_zh": "導數",
      "reason_zh": "固定其他座標後使用單變數導數。"
    },
    {
      "id": "mko-rate-of-change",
      "title_zh": "變化率",
      "reason_zh": "偏導描述單一座標方向的瞬時變化。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "x偏導固定y，y偏導固定x。",
      "梯度只是各座標偏導的有序收集。",
      "有限差分是近似或差商樣本，不自動等於偏導。"
    ],
    "common_misconception_zh": "同時改變兩個座標，或以一個非零h差商聲稱偏導存在。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "偏導存在不推出一般連續性、全微分或混合偏導可交換。",
    "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": "加入梯度、全微分與高階偏導正則性。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-partial-derivative.md 整理；2026-10-13 脈衝 R04（使用者手動接續 2026-10-24 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}