{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261009-r04-derivative-rules-0084",
  "pilot_number": "R0006-03684",
  "number": 3684,
  "family_id": "qf-density-derivative-sum-coefficients",
  "title": {
    "zh-Hant": "第 3684 題：多項式和的完整導數係數",
    "en": "Question 3684: Complete Ascending Derivative Coefficients of a Sum"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261009-r04-derivative-rules-0084",
    "batch_id": "batch-20261007-advance-20261009-r04-derivative-rules-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-derivative-rules"
    ],
    "title_zh": "多項式和的完整導數係數",
    "stem_zh": "全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 8, 10, 3]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。",
    "answer": {
      "kind": "expression",
      "value": "[9, 22, 12]",
      "explanation_zh": "和的導數升冪係數=[9, 22, 12]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "多項式和的完整導數係數"
      ]
    },
    "variant": {
      "family_id": "qf-density-derivative-sum-coefficients",
      "seed": 202610094084,
      "parameters": {
        "f": [
          1,
          8,
          10,
          3
        ],
        "g": [
          -1,
          1,
          1,
          1
        ],
        "operation": "density_derivative_sum_coefficients",
        "domain": "reals",
        "representation": "ascending_powers",
        "semantic_template": "qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,8,10,3]]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-07",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
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    "question_id": "q-20261007-advance-20261009-r04-derivative-rules-0084",
    "batch_id": "batch-20261007-advance-20261009-r04-derivative-rules-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "6cff2b2473aa159faa6440c5b16e7efc7eddad0c9a79d6d40c20797666d2af3e",
      "family_id": "qf-density-derivative-sum-coefficients",
      "seed": 202610094084,
      "semantic_template": "qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,8,10,3]]"
    },
    "fields": {
      "title": "Complete Ascending Derivative Coefficients of a Sum",
      "stem": "On all R, f has ascending coefficients [1, 8, 10, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′.",
      "answer": {
        "kind": "expression",
        "value": "[9, 22, 12]",
        "explanation": "Add coefficientwise, differentiate, and remove trailing zero terms to obtain [9, 22, 12]. Preserve [0] for a zero derivative; this is coefficient identity, not a few sampled values."
      },
      "skills": [
        "Complete Ascending Derivative Coefficients of a Sum"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 8, 10, 3]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。",
        "en": "On all R, f has ascending coefficients [1, 8, 10, 3], constant term first and g has ascending coefficients [-1, 1, 1, 1], constant term first. Give the complete normalized ascending integer coefficients of (f+g)′."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "f ascending coefficients=[1, 8, 10, 3]",
      "after": "real polynomials are differentiable;use coefficient identities",
      "relation": "implies",
      "rule_id": "polynomial_derivative_premise",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第k個係數乘x^k；實多項式可逐項微分，常數導數0。公式恆等與只在一點比較候選是不同主張。",
        "en": "Coefficient k multiplies x^k;real polynomials differentiate termwise and constants have derivative0.Coefficient identities and a candidate comparison at one point are different claims."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "f′ coefficient at degree0 := mul(1, 8)",
      "after": "f′ coefficient at degree0 = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用原次數乘係數，次數再降1；常數項微分為0。",
        "en": "Multiply by the original degree and lower it by1;the constant term differentiates to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f′ coefficient at degree0，不以小數近似替代。",
          "en": "Recompute f′ coefficient at degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "8"
        ],
        "output": "8",
        "label": "f′ coefficient at degree0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "f′ coefficient at degree1 := mul(2, 10)",
      "after": "f′ coefficient at degree1 = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用原次數乘係數，次數再降1；常數項微分為0。",
        "en": "Multiply by the original degree and lower it by1;the constant term differentiates to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f′ coefficient at degree1，不以小數近似替代。",
          "en": "Recompute f′ coefficient at degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "10"
        ],
        "output": "20",
        "label": "f′ coefficient at degree1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "f′ coefficient at degree2 := mul(3, 3)",
      "after": "f′ coefficient at degree2 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用原次數乘係數，次數再降1；常數項微分為0。",
        "en": "Multiply by the original degree and lower it by1;the constant term differentiates to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f′ coefficient at degree2，不以小數近似替代。",
          "en": "Recompute f′ coefficient at degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "3"
        ],
        "output": "9",
        "label": "f′ coefficient at degree2"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "g′ coefficient at degree0 := mul(1, 1)",
      "after": "g′ coefficient at degree0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用原次數乘係數，次數再降1；常數項微分為0。",
        "en": "Multiply by the original degree and lower it by1;the constant term differentiates to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g′ coefficient at degree0，不以小數近似替代。",
          "en": "Recompute g′ coefficient at degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "g′ coefficient at degree0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "g′ coefficient at degree1 := mul(2, 1)",
      "after": "g′ coefficient at degree1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用原次數乘係數，次數再降1；常數項微分為0。",
        "en": "Multiply by the original degree and lower it by1;the constant term differentiates to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g′ coefficient at degree1，不以小數近似替代。",
          "en": "Recompute g′ coefficient at degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "g′ coefficient at degree1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "g′ coefficient at degree2 := mul(3, 1)",
      "after": "g′ coefficient at degree2 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用原次數乘係數，次數再降1；常數項微分為0。",
        "en": "Multiply by the original degree and lower it by1;the constant term differentiates to0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g′ coefficient at degree2，不以小數近似替代。",
          "en": "Recompute g′ coefficient at degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "1"
        ],
        "output": "3",
        "label": "g′ coefficient at degree2"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "sum derivative coefficient0 := sum(8, 1)",
      "after": "sum derivative coefficient0 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按相同次數相加導數係數，缺少的高次項為0。",
        "en": "Add derivative coefficients of the same degree,using0 for a missing high-degree term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum derivative coefficient0，不以小數近似替代。",
          "en": "Recompute sum derivative coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "8",
          "1"
        ],
        "output": "9",
        "label": "sum derivative coefficient0"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "sum derivative coefficient1 := sum(20, 2)",
      "after": "sum derivative coefficient1 = 22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按相同次數相加導數係數，缺少的高次項為0。",
        "en": "Add derivative coefficients of the same degree,using0 for a missing high-degree term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum derivative coefficient1，不以小數近似替代。",
          "en": "Recompute sum derivative coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "2"
        ],
        "output": "22",
        "label": "sum derivative coefficient1"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "sum derivative coefficient2 := sum(9, 3)",
      "after": "sum derivative coefficient2 = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按相同次數相加導數係數，缺少的高次項為0。",
        "en": "Add derivative coefficients of the same degree,using0 for a missing high-degree term."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum derivative coefficient2，不以小數近似替代。",
          "en": "Recompute sum derivative coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "3"
        ],
        "output": "12",
        "label": "sum derivative coefficient2"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "[9, 22, 12]",
      "after": "complete derivative coefficient identity;zero polynomial=[0]",
      "relation": "implies",
      "rule_id": "sum_rule_coefficient_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "和法則給(f+g)′=f′+g′；移除末端零不改多項式，零式保留[0]，不是以幾個點猜完整係數。",
        "en": "The sum rule gives(f+g)′=f′+g′.Removing trailing zeros preserves the polynomial;retain[0] for zero.This is not coefficient inference from a few sample points."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "9",
      "22",
      "12"
    ]
  },
  "conclusion": {
    "canonical_value": "[9, 22, 12]",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
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      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
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    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}