{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261009-r04-derivative-rules-0031",
  "pilot_number": "R0006-03631",
  "number": 3631,
  "family_id": "qf-density-derivative-product-candidate",
  "title": {
    "zh-Hant": "第 3631 題：乘積導數候選值的資格",
    "en": "Question 3631: Eligibility of a Candidate Product-Derivative Value"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261009-r04-derivative-rules-0031",
    "batch_id": "batch-20261007-advance-20261009-r04-derivative-rules-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-derivative-rules"
    ],
    "title_zh": "乘積導數候選值的資格",
    "stem_zh": "全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-7, 6, 1, 2]、g=[2, -1, 1, 3]，x=0，候選整數d=19。d是否等於 (fg)′(x)？",
    "answer": {
      "kind": "short_text",
      "value": "是",
      "explanation_zh": "乘積導數=19/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "乘積導數候選值的資格"
      ]
    },
    "variant": {
      "family_id": "qf-density-derivative-product-candidate",
      "seed": 202610094031,
      "parameters": {
        "f": [
          -7,
          6,
          1,
          2
        ],
        "g": [
          2,
          -1,
          1,
          3
        ],
        "x": 0,
        "candidate": 19,
        "operation": "density_derivative_product_candidate",
        "domain": "reals",
        "representation": "ascending_powers",
        "semantic_template": "qf-density-derivative-product-candidate/[[[-7,6,1,2],[2,-1,1,3]],0,19]"
      }
    },
    "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": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261007-advance-20261009-r04-derivative-rules-0031",
    "batch_id": "batch-20261007-advance-20261009-r04-derivative-rules-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ce14b0edadff561bef2440f37fdc9a8fb33a541a9edbeef6273e1b624498c7d4",
      "family_id": "qf-density-derivative-product-candidate",
      "seed": 202610094031,
      "semantic_template": "qf-density-derivative-product-candidate/[[[-7,6,1,2],[2,-1,1,3]],0,19]"
    },
    "fields": {
      "title": "Eligibility of a Candidate Product-Derivative Value",
      "stem": "On all R, f has ascending coefficients [-7, 6, 1, 2], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=0, is 19 the derivative of fg?",
      "answer": {
        "kind": "short_text",
        "value": "Yes",
        "explanation": "Compare the candidate with the exact derivative of the complete product, giving Yes. The rule is f′g+fg′, not f′g′; retain all resulting terms."
      },
      "skills": [
        "Eligibility of a Candidate Product-Derivative Value"
      ],
      "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=[-7, 6, 1, 2]、g=[2, -1, 1, 3]，x=0，候選整數d=19。d是否等於 (fg)′(x)？",
        "en": "On all R, f has ascending coefficients [-7, 6, 1, 2], constant term first and g has ascending coefficients [2, -1, 1, 3], constant term first. At x=0, is 19 the derivative of fg?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "f ascending coefficients=[-7, 6, 1, 2]",
      "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, 6)",
      "after": "f′ coefficient at degree0 = 6",
      "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",
          "6"
        ],
        "output": "6",
        "label": "f′ coefficient at degree0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "f′ coefficient at degree1 := mul(2, 1)",
      "after": "f′ 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 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",
          "1"
        ],
        "output": "2",
        "label": "f′ coefficient at degree1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "f′ coefficient at degree2 := mul(3, 2)",
      "after": "f′ coefficient at degree2 = 6",
      "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",
          "2"
        ],
        "output": "6",
        "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, 3)",
      "after": "g′ 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 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",
          "3"
        ],
        "output": "9",
        "label": "g′ coefficient at degree2"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "f(x) ascending coefficients=[-7, 6, 1, 2]",
      "after": "evaluate at x=0",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "f(x) Horner multiply before degree2 := mul(2, 0)",
      "after": "f(x) Horner multiply before degree2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f(x) Horner multiply before degree2，不以小數近似替代。",
          "en": "Recompute f(x) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "0"
        ],
        "output": "0",
        "label": "f(x) Horner multiply before degree2"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "f(x) Horner add degree2 := sum(0, 1)",
      "after": "f(x) Horner add degree2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f(x) Horner add degree2，不以小數近似替代。",
          "en": "Recompute f(x) Horner add degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "f(x) Horner add degree2"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "f(x) Horner multiply before degree1 := mul(1, 0)",
      "after": "f(x) Horner multiply before degree1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f(x) Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute f(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "f(x) Horner multiply before degree1"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "f(x) Horner add degree1 := sum(0, 6)",
      "after": "f(x) Horner add degree1 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f(x) Horner add degree1，不以小數近似替代。",
          "en": "Recompute f(x) Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "6"
        ],
        "output": "6",
        "label": "f(x) Horner add degree1"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "f(x) Horner multiply before degree0 := mul(6, 0)",
      "after": "f(x) Horner multiply before degree0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f(x) Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute f(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "0"
        ],
        "output": "0",
        "label": "f(x) Horner multiply before degree0"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "f(x) Horner add degree0 := sum(0, -7)",
      "after": "f(x) Horner add degree0 = -7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f(x) Horner add degree0，不以小數近似替代。",
          "en": "Recompute f(x) Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-7"
        ],
        "output": "-7",
        "label": "f(x) Horner add degree0"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "g(x) ascending coefficients=[2, -1, 1, 3]",
      "after": "evaluate at x=0",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "g(x) Horner multiply before degree2 := mul(3, 0)",
      "after": "g(x) Horner multiply before degree2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g(x) Horner multiply before degree2，不以小數近似替代。",
          "en": "Recompute g(x) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "0"
        ],
        "output": "0",
        "label": "g(x) Horner multiply before degree2"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "g(x) Horner add degree2 := sum(0, 1)",
      "after": "g(x) Horner add degree2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g(x) Horner add degree2，不以小數近似替代。",
          "en": "Recompute g(x) Horner add degree2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "g(x) Horner add degree2"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "g(x) Horner multiply before degree1 := mul(1, 0)",
      "after": "g(x) Horner multiply before degree1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g(x) Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute g(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "g(x) Horner multiply before degree1"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "g(x) Horner add degree1 := sum(0, -1)",
      "after": "g(x) Horner add degree1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g(x) Horner add degree1，不以小數近似替代。",
          "en": "Recompute g(x) Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "-1",
        "label": "g(x) Horner add degree1"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "g(x) Horner multiply before degree0 := mul(-1, 0)",
      "after": "g(x) Horner multiply before degree0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g(x) Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute g(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "0"
        ],
        "output": "0",
        "label": "g(x) Horner multiply before degree0"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "g(x) Horner add degree0 := sum(0, 2)",
      "after": "g(x) Horner add degree0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g(x) Horner add degree0，不以小數近似替代。",
          "en": "Recompute g(x) Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2"
        ],
        "output": "2",
        "label": "g(x) Horner add degree0"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "f′(x) ascending coefficients=[6, 2, 6]",
      "after": "evaluate at x=0",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "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": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "f′(x) Horner multiply before degree1 := mul(6, 0)",
      "after": "f′(x) Horner multiply before degree1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f′(x) Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute f′(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "0"
        ],
        "output": "0",
        "label": "f′(x) Horner multiply before degree1"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "f′(x) Horner add degree1 := sum(0, 2)",
      "after": "f′(x) Horner add degree1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f′(x) Horner add degree1，不以小數近似替代。",
          "en": "Recompute f′(x) Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2"
        ],
        "output": "2",
        "label": "f′(x) Horner add degree1"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "f′(x) Horner multiply before degree0 := mul(2, 0)",
      "after": "f′(x) Horner multiply before degree0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f′(x) Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute f′(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "0"
        ],
        "output": "0",
        "label": "f′(x) Horner multiply before degree0"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "f′(x) Horner add degree0 := sum(0, 6)",
      "after": "f′(x) Horner add degree0 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f′(x) Horner add degree0，不以小數近似替代。",
          "en": "Recompute f′(x) Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "6"
        ],
        "output": "6",
        "label": "f′(x) Horner add degree0"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "g′(x) ascending coefficients=[-1, 2, 9]",
      "after": "evaluate at x=0",
      "relation": "implies",
      "rule_id": "polynomial_evaluation_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "升冪係數用Horner由最高非零項向常數項求值；略去零最高項不改函數，也不改原始係數清單。",
        "en": "Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list."
      },
      "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": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "g′(x) Horner multiply before degree1 := mul(9, 0)",
      "after": "g′(x) Horner multiply before degree1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g′(x) Horner multiply before degree1，不以小數近似替代。",
          "en": "Recompute g′(x) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "0"
        ],
        "output": "0",
        "label": "g′(x) Horner multiply before degree1"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "g′(x) Horner add degree1 := sum(0, 2)",
      "after": "g′(x) Horner add degree1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g′(x) Horner add degree1，不以小數近似替代。",
          "en": "Recompute g′(x) Horner add degree1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "2"
        ],
        "output": "2",
        "label": "g′(x) Horner add degree1"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "g′(x) Horner multiply before degree0 := mul(2, 0)",
      "after": "g′(x) Horner multiply before degree0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘同一代入點。",
        "en": "Multiply by the same evaluation point first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g′(x) Horner multiply before degree0，不以小數近似替代。",
          "en": "Recompute g′(x) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "0"
        ],
        "output": "0",
        "label": "g′(x) Horner multiply before degree0"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "g′(x) Horner add degree0 := sum(0, -1)",
      "after": "g′(x) Horner add degree0 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加此位置的完整係數。",
        "en": "Then add the full coefficient at this position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 g′(x) Horner add degree0，不以小數近似替代。",
          "en": "Recompute g′(x) Horner add degree0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "-1",
        "label": "g′(x) Horner add degree0"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "f′(x)g(x) := mul(6, 2)",
      "after": "f′(x)g(x) = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "乘積法則第一項：只微分f，g保留原值。",
        "en": "First product-rule term:differentiate f and retain g."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f′(x)g(x)，不以小數近似替代。",
          "en": "Recompute f′(x)g(x) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "2"
        ],
        "output": "12",
        "label": "f′(x)g(x)"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "f(x)g′(x) := mul(-7, -1)",
      "after": "f(x)g′(x) = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二項只微分g，不能誤用f′g′。",
        "en": "The second term differentiates only g;do not use f′g′."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 f(x)g′(x)，不以小數近似替代。",
          "en": "Recompute f(x)g′(x) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-7",
          "-1"
        ],
        "output": "7",
        "label": "f(x)g′(x)"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "product derivative := sum(12, 7)",
      "after": "product derivative = 19",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相加兩項得到(fg)′(x)。",
        "en": "Add both terms to obtain(fg)′(x)."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 product derivative，不以小數近似替代。",
          "en": "Recompute product derivative with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "12",
          "7"
        ],
        "output": "19",
        "label": "product derivative"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "candidate=19;actual product derivative=19",
      "after": "是",
      "relation": "implies",
      "rule_id": "product_rule_candidate_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "候選只在此指定x比較；正確微分規則仍依可微多項式前提，而非由單點巧合建立。",
        "en": "Compare the candidate at this specified x;the differentiation rule rests on polynomial differentiability,not a one-point coincidence."
      },
      "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": true,
    "actual_derivative": "19",
    "terms": [
      "12",
      "7"
    ]
  },
  "conclusion": {
    "canonical_value": "是",
    "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."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261007-advance-20261009-r04-derivative-rules-foundations",
      "batch_path": "public/data/questions/batches/batch-20261007-advance-20261009-r04-derivative-rules-foundations/questions.jsonl",
      "batch_sha256": "27c8a3e0551c40adb81570980ba8065daa5c2c723a55d4b4b2750309c8c00b0f",
      "original_line": 31,
      "original_line_sha256": "5ad502c72b2c49ea0467ca907a23667d416a836b4d018da38b1255d5852818d8",
      "question_compact_sha256": "ce14b0edadff561bef2440f37fdc9a8fb33a541a9edbeef6273e1b624498c7d4",
      "english_row_compact_sha256": "115680950a1344105c55f091c8c084cb5db41d089243fb97e156560431d97de9",
      "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261009-r04-derivative-rules-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261009-r04-derivative-rules-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-derivative-rules": {
        "sha256": "4756c1f6226c8cecc330975377ab5d42ed0e461b0fe039c05959fc60ab7c628f",
        "english_sha256": "ed443b25aa7217b775b464078d3230bca85b98df615f3517b6284c5f828ded44",
        "original_path": "public/data/mko/derivative-rules.json",
        "english_path": "public/data/i18n/editions/mko-derivative-rules.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6bac5f83f28cdd9c992a469306bfdf5c5415de17d11847cb9b17b4d13934c7e6",
    "original_candidate_sha256": "6cafaee82c586d9c8e6974973048bc736f9b11063977365077e53da0a49d8e6d",
    "original_candidate_line_sha256": "6cafaee82c586d9c8e6974973048bc736f9b11063977365077e53da0a49d8e6d"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "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."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-004-p7",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "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.",
    "review_receipts": {
      "semantic_review_sha256": "568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d",
      "question_first_witness_sha256": "d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 35,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "2af28e5c9da19f2c9539ba5f8a12535580f885f2f97389d504b22f034b2f4452",
      "arithmetic_steps": 29
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}