{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0001","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[3, 1]、g=[-5, -1]，兩輸入多項式各至多三次；求 (f+g) 在 x=2 的導數值。","answer":{"kind":"short_text","value":"0","explanation_zh":"規則導數值=0。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085001,"parameters":{"f":[3,1],"g":[-5,-1],"x":2,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[\"constant_sum\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0002","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-11, 2, 0, 2]、g=[1, -1, 1, 1]，兩輸入多項式各至多三次；求 (f+g) 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"8","explanation_zh":"規則導數值=8。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085002,"parameters":{"f":[-11,2,0,2],"g":[1,-1,1,1],"x":-1,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"1\",\"1\",\"3\"],\"-1\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0003","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-10, 3, 1, 3]、g=[0, -1, 1, -1]，兩輸入多項式各至多三次；求 (f+g) 在 x=0 的導數值。","answer":{"kind":"short_text","value":"2","explanation_zh":"規則導數值=2。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085003,"parameters":{"f":[-10,3,1,3],"g":[0,-1,1,-1],"x":0,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"2\",\"2\",\"2\"],\"0\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0004","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-9, 4, -1, 4]、g=[-1, -1, 1, 1]，兩輸入多項式各至多三次；求 (f+g) 在 x=1 的導數值。","answer":{"kind":"short_text","value":"18","explanation_zh":"規則導數值=18。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085004,"parameters":{"f":[-9,4,-1,4],"g":[-1,-1,1,1],"x":1,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"3\",\"0\",\"5\"],\"1\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0005","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-8, 5, 0, 1]、g=[-2, -1, 1, -1]，兩輸入多項式各至多三次；求 (f+g) 在 x=2 的導數值。","answer":{"kind":"short_text","value":"8","explanation_zh":"規則導數值=8。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085005,"parameters":{"f":[-8,5,0,1],"g":[-2,-1,1,-1],"x":2,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"4\",\"1\"],\"2\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0006","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-7, 6, 1, 2]、g=[-3, -1, 1, 1]，兩輸入多項式各至多三次；求 (f+g) 在 x=-2 的導數值。","answer":{"kind":"short_text","value":"33","explanation_zh":"規則導數值=33。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085006,"parameters":{"f":[-7,6,1,2],"g":[-3,-1,1,1],"x":-2,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"5\",\"2\",\"3\"],\"-2\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0007","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-6, 7, -1, 3]、g=[-4, -1, 1, -1]，兩輸入多項式各至多三次；求 (f+g) 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"12","explanation_zh":"規則導數值=12。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085007,"parameters":{"f":[-6,7,-1,3],"g":[-4,-1,1,-1],"x":-1,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"6\",\"0\",\"2\"],\"-1\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0008","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-5, 8, 0, 4]、g=[-5, -1, 1, 1]，兩輸入多項式各至多三次；求 (f+g) 在 x=0 的導數值。","answer":{"kind":"short_text","value":"7","explanation_zh":"規則導數值=7。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085008,"parameters":{"f":[-5,8,0,4],"g":[-5,-1,1,1],"x":0,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"7\",\"1\",\"5\"],\"0\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0009","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-4, 9, 1, 1]、g=[-6, -1, 1, -1]，兩輸入多項式各至多三次；求 (f+g) 在 x=1 的導數值。","answer":{"kind":"short_text","value":"12","explanation_zh":"規則導數值=12。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085009,"parameters":{"f":[-4,9,1,1],"g":[-6,-1,1,-1],"x":1,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"8\",\"2\"],\"1\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0010","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-3, 10, -1, 2]、g=[-7, -1, 1, 1]，兩輸入多項式各至多三次；求 (f+g) 在 x=2 的導數值。","answer":{"kind":"short_text","value":"45","explanation_zh":"規則導數值=45。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085010,"parameters":{"f":[-3,10,-1,2],"g":[-7,-1,1,1],"x":2,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"9\",\"0\",\"3\"],\"2\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0011","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-2, 11, 0, 3]、g=[-8, -1, 1, -1]，兩輸入多項式各至多三次；求 (f+g) 在 x=-2 的導數值。","answer":{"kind":"short_text","value":"30","explanation_zh":"規則導數值=30。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085011,"parameters":{"f":[-2,11,0,3],"g":[-8,-1,1,-1],"x":-2,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"10\",\"1\",\"2\"],\"-2\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0012","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-1, 12, 1, 4]、g=[-9, -1, 1, 1]，兩輸入多項式各至多三次；求 (f+g) 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"22","explanation_zh":"規則導數值=22。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085012,"parameters":{"f":[-1,12,1,4],"g":[-9,-1,1,1],"x":-1,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"11\",\"2\",\"5\"],\"-1\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0013","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[0, 13, -1, 1]、g=[-10, -1, 1, -1]，兩輸入多項式各至多三次；求 (f+g) 在 x=0 的導數值。","answer":{"kind":"short_text","value":"12","explanation_zh":"規則導數值=12。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085013,"parameters":{"f":[0,13,-1,1],"g":[-10,-1,1,-1],"x":0,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"12\"],\"point_irrelevant\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0014","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[1, 14, 0, 2]、g=[-11, -1, 1, 1]，兩輸入多項式各至多三次；求 (f+g) 在 x=1 的導數值。","answer":{"kind":"short_text","value":"24","explanation_zh":"規則導數值=24。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085014,"parameters":{"f":[1,14,0,2],"g":[-11,-1,1,1],"x":1,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"13\",\"1\",\"3\"],\"1\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0015","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[2, 15, 1, 3]、g=[-12, -1, 1, -1]，兩輸入多項式各至多三次；求 (f+g) 在 x=2 的導數值。","answer":{"kind":"short_text","value":"46","explanation_zh":"規則導數值=46。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085015,"parameters":{"f":[2,15,1,3],"g":[-12,-1,1,-1],"x":2,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"14\",\"2\",\"2\"],\"2\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0016","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[3, 16, -1, 4]、g=[-13, -1, 1, 1]，兩輸入多項式各至多三次；求 (f+g) 在 x=-2 的導數值。","answer":{"kind":"short_text","value":"75","explanation_zh":"規則導數值=75。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085016,"parameters":{"f":[3,16,-1,4],"g":[-13,-1,1,1],"x":-2,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"15\",\"0\",\"5\"],\"-2\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0017","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[4, 17, 0, 1]、g=[-14, -1, 1, -1]，兩輸入多項式各至多三次；求 (f+g) 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"14","explanation_zh":"規則導數值=14。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085017,"parameters":{"f":[4,17,0,1],"g":[-14,-1,1,-1],"x":-1,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"16\",\"1\"],\"-1\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0018","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[5, 18, 1, 2]、g=[-15, -1, 1, 1]，兩輸入多項式各至多三次；求 (f+g) 在 x=0 的導數值。","answer":{"kind":"short_text","value":"17","explanation_zh":"規則導數值=17。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085018,"parameters":{"f":[5,18,1,2],"g":[-15,-1,1,1],"x":0,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"17\",\"2\",\"3\"],\"0\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0019","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[6, 19, -1, 3]、g=[-16, -1, 1, -1]，兩輸入多項式各至多三次；求 (f+g) 在 x=1 的導數值。","answer":{"kind":"short_text","value":"24","explanation_zh":"規則導數值=24。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085019,"parameters":{"f":[6,19,-1,3],"g":[-16,-1,1,-1],"x":1,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"18\",\"0\",\"2\"],\"1\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0020","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[7, 20, 0, 4]、g=[-17, -1, 1, 1]，兩輸入多項式各至多三次；求 (f+g) 在 x=2 的導數值。","answer":{"kind":"short_text","value":"83","explanation_zh":"規則導數值=83。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085020,"parameters":{"f":[7,20,0,4],"g":[-17,-1,1,1],"x":2,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"19\",\"1\",\"5\"],\"2\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0021","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[8, 21, 1, 1]、g=[-18, -1, 1, -1]，兩輸入多項式各至多三次；求 (f+g) 在 x=-2 的導數值。","answer":{"kind":"short_text","value":"12","explanation_zh":"規則導數值=12。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085021,"parameters":{"f":[8,21,1,1],"g":[-18,-1,1,-1],"x":-2,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"20\",\"2\"],\"-2\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0022","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[9, 22, -1, 2]、g=[-19, -1, 1, 1]，兩輸入多項式各至多三次；求 (f+g) 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"30","explanation_zh":"規則導數值=30。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085022,"parameters":{"f":[9,22,-1,2],"g":[-19,-1,1,1],"x":-1,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"21\",\"0\",\"3\"],\"-1\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0023","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[10, 23, 0, 3]、g=[-20, -1, 1, -1]，兩輸入多項式各至多三次；求 (f+g) 在 x=0 的導數值。","answer":{"kind":"short_text","value":"22","explanation_zh":"規則導數值=22。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085023,"parameters":{"f":[10,23,0,3],"g":[-20,-1,1,-1],"x":0,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"22\",\"1\",\"2\"],\"0\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0024","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[11, 24, 1, 4]、g=[-21, -1, 1, 1]，兩輸入多項式各至多三次；求 (f+g) 在 x=1 的導數值。","answer":{"kind":"short_text","value":"42","explanation_zh":"規則導數值=42。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085024,"parameters":{"f":[11,24,1,4],"g":[-21,-1,1,1],"x":1,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"23\",\"2\",\"5\"],\"1\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0025","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[12, 25, -1, 1]、g=[-22, -1, 1, -1]，兩輸入多項式各至多三次；求 (f+g) 在 x=2 的導數值。","answer":{"kind":"short_text","value":"24","explanation_zh":"規則導數值=24。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的導數"]},"variant":{"family_id":"qf-derivative-rule-sum","seed":202610085025,"parameters":{"f":[12,25,-1,1],"g":[-22,-1,1,-1],"x":2,"operation":"derivative_sum","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-sum/{\"inputs\":[[\"24\"],\"point_irrelevant\"],\"operation\":\"derivative_sum\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0026","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[1, 1]，兩輸入多項式各至多三次；求 0f 在 x=-2 的導數值。","answer":{"kind":"short_text","value":"0","explanation_zh":"規則導數值=0。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085026,"parameters":{"f":[1,1],"c":0,"x":-2,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[\"zero_derivative\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0027","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-11, 2, 0, 2]，兩輸入多項式各至多三次；求 -2f 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"-16","explanation_zh":"規則導數值=-16。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085027,"parameters":{"f":[-11,2,0,2],"c":-2,"x":-1,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"2\",\"0\",\"2\"],\"-2\",\"-1\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0028","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-10, 3, 1, 3]，兩輸入多項式各至多三次；求 3f 在 x=0 的導數值。","answer":{"kind":"short_text","value":"9","explanation_zh":"規則導數值=9。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085028,"parameters":{"f":[-10,3,1,3],"c":3,"x":0,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"3\",\"1\",\"3\"],\"3\",\"0\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0029","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-9, 4, -1, 4]，兩輸入多項式各至多三次；求 -4f 在 x=1 的導數值。","answer":{"kind":"short_text","value":"-56","explanation_zh":"規則導數值=-56。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085029,"parameters":{"f":[-9,4,-1,4],"c":-4,"x":1,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"4\",\"-1\",\"4\"],\"-4\",\"1\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0030","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-8, 5, 0, 1]，兩輸入多項式各至多三次；求 1f 在 x=2 的導數值。","answer":{"kind":"short_text","value":"17","explanation_zh":"規則導數值=17。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085030,"parameters":{"f":[-8,5,0,1],"c":1,"x":2,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"5\",\"0\",\"1\"],\"1\",\"2\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0031","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-7, 6, 1, 2]，兩輸入多項式各至多三次；求 -2f 在 x=-2 的導數值。","answer":{"kind":"short_text","value":"-52","explanation_zh":"規則導數值=-52。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085031,"parameters":{"f":[-7,6,1,2],"c":-2,"x":-2,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"6\",\"1\",\"2\"],\"-2\",\"-2\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0032","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-6, 7, -1, 3]，兩輸入多項式各至多三次；求 3f 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"54","explanation_zh":"規則導數值=54。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085032,"parameters":{"f":[-6,7,-1,3],"c":3,"x":-1,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"7\",\"-1\",\"3\"],\"3\",\"-1\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0033","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-5, 8, 0, 4]，兩輸入多項式各至多三次；求 -4f 在 x=0 的導數值。","answer":{"kind":"short_text","value":"-32","explanation_zh":"規則導數值=-32。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085033,"parameters":{"f":[-5,8,0,4],"c":-4,"x":0,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"8\",\"0\",\"4\"],\"-4\",\"0\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0034","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-4, 9, 1, 1]，兩輸入多項式各至多三次；求 1f 在 x=1 的導數值。","answer":{"kind":"short_text","value":"14","explanation_zh":"規則導數值=14。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085034,"parameters":{"f":[-4,9,1,1],"c":1,"x":1,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"9\",\"1\",\"1\"],\"1\",\"1\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0035","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-3, 10, -1, 2]，兩輸入多項式各至多三次；求 -2f 在 x=2 的導數值。","answer":{"kind":"short_text","value":"-60","explanation_zh":"規則導數值=-60。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085035,"parameters":{"f":[-3,10,-1,2],"c":-2,"x":2,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"10\",\"-1\",\"2\"],\"-2\",\"2\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0036","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-2, 11, 0, 3]，兩輸入多項式各至多三次；求 3f 在 x=-2 的導數值。","answer":{"kind":"short_text","value":"141","explanation_zh":"規則導數值=141。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085036,"parameters":{"f":[-2,11,0,3],"c":3,"x":-2,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"11\",\"0\",\"3\"],\"3\",\"-2\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0037","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-1, 12, 1, 4]，兩輸入多項式各至多三次；求 -4f 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"-88","explanation_zh":"規則導數值=-88。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085037,"parameters":{"f":[-1,12,1,4],"c":-4,"x":-1,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"12\",\"1\",\"4\"],\"-4\",\"-1\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0038","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[0, 13, -1, 1]，兩輸入多項式各至多三次；求 1f 在 x=0 的導數值。","answer":{"kind":"short_text","value":"13","explanation_zh":"規則導數值=13。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085038,"parameters":{"f":[0,13,-1,1],"c":1,"x":0,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"13\",\"-1\",\"1\"],\"1\",\"0\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0039","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[1, 14, 0, 2]，兩輸入多項式各至多三次；求 -2f 在 x=1 的導數值。","answer":{"kind":"short_text","value":"-40","explanation_zh":"規則導數值=-40。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085039,"parameters":{"f":[1,14,0,2],"c":-2,"x":1,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"14\",\"0\",\"2\"],\"-2\",\"1\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0040","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[2, 15, 1, 3]，兩輸入多項式各至多三次；求 3f 在 x=2 的導數值。","answer":{"kind":"short_text","value":"165","explanation_zh":"規則導數值=165。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085040,"parameters":{"f":[2,15,1,3],"c":3,"x":2,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"15\",\"1\",\"3\"],\"3\",\"2\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0041","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[3, 16, -1, 4]，兩輸入多項式各至多三次；求 -4f 在 x=-2 的導數值。","answer":{"kind":"short_text","value":"-272","explanation_zh":"規則導數值=-272。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085041,"parameters":{"f":[3,16,-1,4],"c":-4,"x":-2,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"16\",\"-1\",\"4\"],\"-4\",\"-2\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0042","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[4, 17, 0, 1]，兩輸入多項式各至多三次；求 1f 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"20","explanation_zh":"規則導數值=20。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085042,"parameters":{"f":[4,17,0,1],"c":1,"x":-1,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"17\",\"0\",\"1\"],\"1\",\"-1\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0043","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[5, 18, 1, 2]，兩輸入多項式各至多三次；求 -2f 在 x=0 的導數值。","answer":{"kind":"short_text","value":"-36","explanation_zh":"規則導數值=-36。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085043,"parameters":{"f":[5,18,1,2],"c":-2,"x":0,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"18\",\"1\",\"2\"],\"-2\",\"0\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0044","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[6, 19, -1, 3]，兩輸入多項式各至多三次；求 3f 在 x=1 的導數值。","answer":{"kind":"short_text","value":"78","explanation_zh":"規則導數值=78。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085044,"parameters":{"f":[6,19,-1,3],"c":3,"x":1,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"19\",\"-1\",\"3\"],\"3\",\"1\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0045","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[7, 20, 0, 4]，兩輸入多項式各至多三次；求 -4f 在 x=2 的導數值。","answer":{"kind":"short_text","value":"-272","explanation_zh":"規則導數值=-272。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085045,"parameters":{"f":[7,20,0,4],"c":-4,"x":2,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"20\",\"0\",\"4\"],\"-4\",\"2\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0046","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[8, 21, 1, 1]，兩輸入多項式各至多三次；求 1f 在 x=-2 的導數值。","answer":{"kind":"short_text","value":"29","explanation_zh":"規則導數值=29。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085046,"parameters":{"f":[8,21,1,1],"c":1,"x":-2,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"21\",\"1\",\"1\"],\"1\",\"-2\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0047","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[9, 22, -1, 2]，兩輸入多項式各至多三次；求 -2f 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"-60","explanation_zh":"規則導數值=-60。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085047,"parameters":{"f":[9,22,-1,2],"c":-2,"x":-1,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"22\",\"-1\",\"2\"],\"-2\",\"-1\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0048","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[10, 23, 0, 3]，兩輸入多項式各至多三次；求 3f 在 x=0 的導數值。","answer":{"kind":"short_text","value":"69","explanation_zh":"規則導數值=69。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085048,"parameters":{"f":[10,23,0,3],"c":3,"x":0,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"23\",\"0\",\"3\"],\"3\",\"0\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0049","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[11, 24, 1, 4]，兩輸入多項式各至多三次；求 -4f 在 x=1 的導數值。","answer":{"kind":"short_text","value":"-152","explanation_zh":"規則導數值=-152。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085049,"parameters":{"f":[11,24,1,4],"c":-4,"x":1,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"24\",\"1\",\"4\"],\"-4\",\"1\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0050","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式純量倍的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[12, 25, -1, 1]，兩輸入多項式各至多三次；求 1f 在 x=2 的導數值。","answer":{"kind":"short_text","value":"33","explanation_zh":"規則導數值=33。只對已明示多項式使用和／純量法則，不處理任意未宣告可微函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式純量倍的導數"]},"variant":{"family_id":"qf-derivative-rule-scalar","seed":202610085050,"parameters":{"f":[12,25,-1,1],"c":1,"x":2,"operation":"derivative_scalar","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-scalar/{\"inputs\":[[\"25\",\"-1\",\"1\"],\"1\",\"2\"],\"operation\":\"derivative_scalar\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0051","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[3]、g=[-2]，兩輸入多項式各至多三次；求 fg 在 x=1 的導數值。","answer":{"kind":"short_text","value":"0","explanation_zh":"規則導數值=0。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085051,"parameters":{"f":[3],"g":[-2],"x":1,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[\"constant_product\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0052","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-11, 2, 0, 2]、g=[2, 0, 1, 3]，兩輸入多項式各至多三次；求 fg 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"-105","explanation_zh":"規則導數值=-105。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085052,"parameters":{"f":[-11,2,0,2],"g":[2,0,1,3],"x":-1,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-11\",\"2\",\"0\",\"2\"],[\"2\",\"0\",\"1\",\"3\"]],\"-1\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0053","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-10, 3, 1, 3]、g=[2, 1, 1, 4]，兩輸入多項式各至多三次；求 fg 在 x=0 的導數值。","answer":{"kind":"short_text","value":"-4","explanation_zh":"規則導數值=-4。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085053,"parameters":{"f":[-10,3,1,3],"g":[2,1,1,4],"x":0,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-10\",\"3\",\"1\",\"3\"],[\"2\",\"1\",\"1\",\"4\"]],\"0\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0054","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-9, 4, -1, 4]、g=[2, 2, 1, 2]，兩輸入多項式各至多三次；求 fg 在 x=1 的導數值。","answer":{"kind":"short_text","value":"78","explanation_zh":"規則導數值=78。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085054,"parameters":{"f":[-9,4,-1,4],"g":[2,2,1,2],"x":1,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-9\",\"4\",\"-1\",\"4\"],[\"2\",\"2\",\"1\",\"2\"]],\"1\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0055","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-8, 5, 0, 1]、g=[2, -1, 1, 3]，兩輸入多項式各至多三次；求 fg 在 x=2 的導數值。","answer":{"kind":"short_text","value":"866","explanation_zh":"規則導數值=866。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085055,"parameters":{"f":[-8,5,0,1],"g":[2,-1,1,3],"x":2,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-8\",\"5\",\"0\",\"1\"],[\"2\",\"-1\",\"1\",\"3\"]],\"2\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0056","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-7, 6, 1, 2]、g=[2, 0, 1, 4]，兩輸入多項式各至多三次；求 fg 在 x=-2 的導數值。","answer":{"kind":"short_text","value":"-2040","explanation_zh":"規則導數值=-2040。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085056,"parameters":{"f":[-7,6,1,2],"g":[2,0,1,4],"x":-2,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-7\",\"6\",\"1\",\"2\"],[\"2\",\"0\",\"1\",\"4\"]],\"-2\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0057","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-6, 7, -1, 3]、g=[2, 1, 1, 2]，兩輸入多項式各至多三次；求 fg 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"-85","explanation_zh":"規則導數值=-85。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085057,"parameters":{"f":[-6,7,-1,3],"g":[2,1,1,2],"x":-1,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-6\",\"7\",\"-1\",\"3\"],[\"2\",\"1\",\"1\",\"2\"]],\"-1\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0058","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-5, 8, 0, 4]、g=[2, 2, 1, 3]，兩輸入多項式各至多三次；求 fg 在 x=0 的導數值。","answer":{"kind":"short_text","value":"6","explanation_zh":"規則導數值=6。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085058,"parameters":{"f":[-5,8,0,4],"g":[2,2,1,3],"x":0,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-5\",\"8\",\"0\",\"4\"],[\"2\",\"2\",\"1\",\"3\"]],\"0\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0059","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-4, 9, 1, 1]、g=[2, -1, 1, 4]，兩輸入多項式各至多三次；求 fg 在 x=1 的導數值。","answer":{"kind":"short_text","value":"175","explanation_zh":"規則導數值=175。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085059,"parameters":{"f":[-4,9,1,1],"g":[2,-1,1,4],"x":1,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-4\",\"9\",\"1\",\"1\"],[\"2\",\"-1\",\"1\",\"4\"]],\"1\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0060","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-3, 10, -1, 2]、g=[2, 0, 1, 2]，兩輸入多項式各至多三次；求 fg 在 x=2 的導數值。","answer":{"kind":"short_text","value":"1472","explanation_zh":"規則導數值=1472。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085060,"parameters":{"f":[-3,10,-1,2],"g":[2,0,1,2],"x":2,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-3\",\"10\",\"-1\",\"2\"],[\"2\",\"0\",\"1\",\"2\"]],\"2\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0061","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-2, 11, 0, 3]、g=[2, 1, 1, 3]，兩輸入多項式各至多三次；求 fg 在 x=-2 的導數值。","answer":{"kind":"short_text","value":"-2524","explanation_zh":"規則導數值=-2524。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085061,"parameters":{"f":[-2,11,0,3],"g":[2,1,1,3],"x":-2,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-2\",\"11\",\"0\",\"3\"],[\"2\",\"1\",\"1\",\"3\"]],\"-2\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0062","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-1, 12, 1, 4]、g=[2, 2, 1, 4]，兩輸入多項式各至多三次；求 fg 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"-258","explanation_zh":"規則導數值=-258。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085062,"parameters":{"f":[-1,12,1,4],"g":[2,2,1,4],"x":-1,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"-1\",\"12\",\"1\",\"4\"],[\"2\",\"2\",\"1\",\"4\"]],\"-1\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0063","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[0, 13, -1, 1]、g=[2, -1, 1, 2]，兩輸入多項式各至多三次；求 fg 在 x=0 的導數值。","answer":{"kind":"short_text","value":"26","explanation_zh":"規則導數值=26。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085063,"parameters":{"f":[0,13,-1,1],"g":[2,-1,1,2],"x":0,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"0\",\"13\",\"-1\",\"1\"],[\"2\",\"-1\",\"1\",\"2\"]],\"0\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0064","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[1, 14, 0, 2]、g=[2, 0, 1, 3]，兩輸入多項式各至多三次；求 fg 在 x=1 的導數值。","answer":{"kind":"short_text","value":"307","explanation_zh":"規則導數值=307。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085064,"parameters":{"f":[1,14,0,2],"g":[2,0,1,3],"x":1,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"1\",\"14\",\"0\",\"2\"],[\"2\",\"0\",\"1\",\"3\"]],\"1\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0065","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[2, 15, 1, 3]、g=[2, 1, 1, 4]，兩輸入多項式各至多三次；求 fg 在 x=2 的導數值。","answer":{"kind":"short_text","value":"5380","explanation_zh":"規則導數值=5380。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085065,"parameters":{"f":[2,15,1,3],"g":[2,1,1,4],"x":2,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"1\",\"1\",\"4\"],[\"2\",\"15\",\"1\",\"3\"]],\"2\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0066","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[3, 16, -1, 4]、g=[2, 2, 1, 2]，兩輸入多項式各至多三次；求 fg 在 x=-2 的導數值。","answer":{"kind":"short_text","value":"-2382","explanation_zh":"規則導數值=-2382。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085066,"parameters":{"f":[3,16,-1,4],"g":[2,2,1,2],"x":-2,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"2\",\"1\",\"2\"],[\"3\",\"16\",\"-1\",\"4\"]],\"-2\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0067","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[4, 17, 0, 1]、g=[2, -1, 1, 3]，兩輸入多項式各至多三次；求 fg 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"-64","explanation_zh":"規則導數值=-64。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085067,"parameters":{"f":[4,17,0,1],"g":[2,-1,1,3],"x":-1,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"-1\",\"1\",\"3\"],[\"4\",\"17\",\"0\",\"1\"]],\"-1\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0068","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[5, 18, 1, 2]、g=[2, 0, 1, 4]，兩輸入多項式各至多三次；求 fg 在 x=0 的導數值。","answer":{"kind":"short_text","value":"36","explanation_zh":"規則導數值=36。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085068,"parameters":{"f":[5,18,1,2],"g":[2,0,1,4],"x":0,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"0\",\"1\",\"4\"],[\"5\",\"18\",\"1\",\"2\"]],\"0\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0069","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[6, 19, -1, 3]、g=[2, 1, 1, 2]，兩輸入多項式各至多三次；求 fg 在 x=1 的導數值。","answer":{"kind":"short_text","value":"399","explanation_zh":"規則導數值=399。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085069,"parameters":{"f":[6,19,-1,3],"g":[2,1,1,2],"x":1,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"1\",\"1\",\"2\"],[\"6\",\"19\",\"-1\",\"3\"]],\"1\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0070","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[7, 20, 0, 4]、g=[2, 2, 1, 3]，兩輸入多項式各至多三次；求 fg 在 x=2 的導數值。","answer":{"kind":"short_text","value":"5630","explanation_zh":"規則導數值=5630。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085070,"parameters":{"f":[7,20,0,4],"g":[2,2,1,3],"x":2,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"2\",\"1\",\"3\"],[\"7\",\"20\",\"0\",\"4\"]],\"2\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0071","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[8, 21, 1, 1]、g=[2, -1, 1, 4]，兩輸入多項式各至多三次；求 fg 在 x=-2 的導數值。","answer":{"kind":"short_text","value":"-2330","explanation_zh":"規則導數值=-2330。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085071,"parameters":{"f":[8,21,1,1],"g":[2,-1,1,4],"x":-2,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"-1\",\"1\",\"4\"],[\"8\",\"21\",\"1\",\"1\"]],\"-2\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0072","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[9, 22, -1, 2]、g=[2, 0, 1, 2]，兩輸入多項式各至多三次；求 fg 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"-34","explanation_zh":"規則導數值=-34。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085072,"parameters":{"f":[9,22,-1,2],"g":[2,0,1,2],"x":-1,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"2\",\"0\",\"1\",\"2\"],[\"9\",\"22\",\"-1\",\"2\"]],\"-1\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0073","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[10, 23, 0, 3]、g=[2, 1, 1, 3]，兩輸入多項式各至多三次；求 fg 在 x=0 的導數值。","answer":{"kind":"short_text","value":"56","explanation_zh":"規則導數值=56。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085073,"parameters":{"f":[10,23,0,3],"g":[2,1,1,3],"x":0,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"10\",\"23\",\"0\",\"3\"],[\"2\",\"1\",\"1\",\"3\"]],\"0\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0074","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[11, 24, 1, 4]、g=[2, 2, 1, 4]，兩輸入多項式各至多三次；求 fg 在 x=1 的導數值。","answer":{"kind":"short_text","value":"982","explanation_zh":"規則導數值=982。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085074,"parameters":{"f":[11,24,1,4],"g":[2,2,1,4],"x":1,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"11\",\"24\",\"1\",\"4\"],[\"2\",\"2\",\"1\",\"4\"]],\"1\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0075","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式乘積的導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[12, 25, -1, 1]、g=[2, -1, 1, 2]，兩輸入多項式各至多三次；求 fg 在 x=2 的導數值。","answer":{"kind":"short_text","value":"2442","explanation_zh":"規則導數值=2442。產生端用 f′g+fg′，不把乘積導數誤當兩導數乘積。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式乘積的導數"]},"variant":{"family_id":"qf-derivative-rule-product","seed":202610085075,"parameters":{"f":[12,25,-1,1],"g":[2,-1,1,2],"x":2,"operation":"derivative_product","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-product/{\"inputs\":[[[\"12\",\"25\",\"-1\",\"1\"],[\"2\",\"-1\",\"1\",\"2\"]],\"2\"],\"operation\":\"derivative_product\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0076","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[1, 2, 3]、g=[4]，兩輸入多項式各至多三次；求 f∘g 在 x=2 的導數值。","answer":{"kind":"short_text","value":"0","explanation_zh":"規則導數值=0。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085076,"parameters":{"f":[1,2,3],"g":[4],"x":2,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[\"constant_composite\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0077","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-11, 2, 1, 1]、g=[-4, 1, 2, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=0 的導數值。","answer":{"kind":"short_text","value":"42","explanation_zh":"規則導數值=42。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085077,"parameters":{"f":[-11,2,1,1],"g":[-4,1,2,1],"x":0,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"2\",\"1\",\"1\"],[\"-4\",\"1\",\"2\",\"1\"],\"0\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0078","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-10, 3, 2, 1]、g=[-3, 1, 3, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=1 的導數值。","answer":{"kind":"short_text","value":"230","explanation_zh":"規則導數值=230。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085078,"parameters":{"f":[-10,3,2,1],"g":[-3,1,3,1],"x":1,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"3\",\"2\",\"1\"],[\"-3\",\"1\",\"3\",\"1\"],\"1\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0079","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-9, 4, 0, 1]、g=[-2, 1, 1, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=2 的導數值。","answer":{"kind":"short_text","value":"7412","explanation_zh":"規則導數值=7412。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085079,"parameters":{"f":[-9,4,0,1],"g":[-2,1,1,1],"x":2,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"4\",\"0\",\"1\"],[\"-2\",\"1\",\"1\",\"1\"],\"2\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0080","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-8, 1, 1, 1]、g=[-1, 1, 2, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"0","explanation_zh":"規則導數值=0。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085080,"parameters":{"f":[-8,1,1,1],"g":[-1,1,2,1],"x":-1,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"1\",\"1\",\"1\"],[\"-1\",\"1\",\"2\",\"1\"],\"-1\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0081","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-7, 2, 2, 1]、g=[0, 1, 3, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=0 的導數值。","answer":{"kind":"short_text","value":"2","explanation_zh":"規則導數值=2。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085081,"parameters":{"f":[-7,2,2,1],"g":[0,1,3,1],"x":0,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"2\",\"2\",\"1\"],[\"0\",\"1\",\"3\",\"1\"],\"0\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0082","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-6, 3, 0, 1]、g=[1, 1, 1, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=1 的導數值。","answer":{"kind":"short_text","value":"306","explanation_zh":"規則導數值=306。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085082,"parameters":{"f":[-6,3,0,1],"g":[1,1,1,1],"x":1,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"3\",\"0\",\"1\"],[\"1\",\"1\",\"1\",\"1\"],\"1\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0083","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-5, 4, 1, 1]、g=[2, 1, 2, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=2 的導數值。","answer":{"kind":"short_text","value":"26124","explanation_zh":"規則導數值=26124。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085083,"parameters":{"f":[-5,4,1,1],"g":[2,1,2,1],"x":2,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"4\",\"1\",\"1\"],[\"2\",\"1\",\"2\",\"1\"],\"2\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0084","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-4, 1, 2, 1]、g=[3, 1, 3, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"-130","explanation_zh":"規則導數值=-130。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085084,"parameters":{"f":[-4,1,2,1],"g":[3,1,3,1],"x":-1,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"1\",\"2\",\"1\"],[\"3\",\"1\",\"3\",\"1\"],\"-1\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0085","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-3, 2, 0, 1]、g=[4, 1, 1, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=0 的導數值。","answer":{"kind":"short_text","value":"50","explanation_zh":"規則導數值=50。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085085,"parameters":{"f":[-3,2,0,1],"g":[4,1,1,1],"x":0,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"2\",\"0\",\"1\"],[\"4\",\"1\",\"1\",\"1\"],\"0\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0086","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-2, 3, 1, 1]、g=[5, 1, 2, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=1 的導數值。","answer":{"kind":"short_text","value":"2112","explanation_zh":"規則導數值=2112。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085086,"parameters":{"f":[-2,3,1,1],"g":[5,1,2,1],"x":1,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"3\",\"1\",\"1\"],[\"5\",\"1\",\"2\",\"1\"],\"1\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0087","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[-1, 4, 2, 1]、g=[6, 1, 3, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=2 的導數值。","answer":{"kind":"short_text","value":"61700","explanation_zh":"規則導數值=61700。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085087,"parameters":{"f":[-1,4,2,1],"g":[6,1,3,1],"x":2,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"4\",\"2\",\"1\"],[\"6\",\"1\",\"3\",\"1\"],\"2\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0088","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[0, 1, 0, 1]、g=[7, 1, 1, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"218","explanation_zh":"規則導數值=218。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085088,"parameters":{"f":[0,1,0,1],"g":[7,1,1,1],"x":-1,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"1\",\"0\",\"1\"],[\"7\",\"1\",\"1\",\"1\"],\"-1\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0089","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[1, 2, 1, 1]、g=[8, 1, 2, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=0 的導數值。","answer":{"kind":"short_text","value":"210","explanation_zh":"規則導數值=210。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085089,"parameters":{"f":[1,2,1,1],"g":[8,1,2,1],"x":0,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"2\",\"1\",\"1\"],[\"8\",\"1\",\"2\",\"1\"],\"0\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0090","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[2, 3, 2, 1]、g=[9, 1, 3, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=1 的導數值。","answer":{"kind":"short_text","value":"6470","explanation_zh":"規則導數值=6470。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085090,"parameters":{"f":[2,3,2,1],"g":[9,1,3,1],"x":1,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"3\",\"2\",\"1\"],[\"9\",\"1\",\"3\",\"1\"],\"1\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0091","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[3, 4, 0, 1]、g=[10, 1, 1, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=2 的導數值。","answer":{"kind":"short_text","value":"29444","explanation_zh":"規則導數值=29444。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085091,"parameters":{"f":[3,4,0,1],"g":[10,1,1,1],"x":2,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"4\",\"0\",\"1\"],[\"10\",\"1\",\"1\",\"1\"],\"2\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0092","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[4, 1, 1, 1]、g=[11, 1, 2, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"0","explanation_zh":"規則導數值=0。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085092,"parameters":{"f":[4,1,1,1],"g":[11,1,2,1],"x":-1,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"1\",\"1\",\"1\"],[\"11\",\"1\",\"2\",\"1\"],\"-1\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0093","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[5, 2, 2, 1]、g=[12, 1, 3, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=0 的導數值。","answer":{"kind":"short_text","value":"482","explanation_zh":"規則導數值=482。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085093,"parameters":{"f":[5,2,2,1],"g":[12,1,3,1],"x":0,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"2\",\"2\",\"1\"],[\"12\",\"1\",\"3\",\"1\"],\"0\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0094","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[6, 3, 0, 1]、g=[13, 1, 1, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=1 的導數值。","answer":{"kind":"short_text","value":"4626","explanation_zh":"規則導數值=4626。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085094,"parameters":{"f":[6,3,0,1],"g":[13,1,1,1],"x":1,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"3\",\"0\",\"1\"],[\"13\",\"1\",\"1\",\"1\"],\"1\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0095","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[7, 4, 1, 1]、g=[14, 1, 2, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=2 的導數值。","answer":{"kind":"short_text","value":"65940","explanation_zh":"規則導數值=65940。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085095,"parameters":{"f":[7,4,1,1],"g":[14,1,2,1],"x":2,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"4\",\"1\",\"1\"],[\"14\",\"1\",\"2\",\"1\"],\"2\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0096","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[8, 1, 2, 1]、g=[15, 1, 3, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"-1666","explanation_zh":"規則導數值=-1666。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085096,"parameters":{"f":[8,1,2,1],"g":[15,1,3,1],"x":-1,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"1\",\"2\",\"1\"],[\"15\",\"1\",\"3\",\"1\"],\"-1\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0097","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[9, 2, 0, 1]、g=[16, 1, 1, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=0 的導數值。","answer":{"kind":"short_text","value":"770","explanation_zh":"規則導數值=770。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085097,"parameters":{"f":[9,2,0,1],"g":[16,1,1,1],"x":0,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"2\",\"0\",\"1\"],[\"16\",\"1\",\"1\",\"1\"],\"0\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0098","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[10, 3, 1, 1]、g=[17, 1, 2, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=1 的導數值。","answer":{"kind":"short_text","value":"10944","explanation_zh":"規則導數值=10944。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085098,"parameters":{"f":[10,3,1,1],"g":[17,1,2,1],"x":1,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"3\",\"1\",\"1\"],[\"17\",\"1\",\"2\",\"1\"],\"1\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0099","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[11, 4, 2, 1]、g=[18, 1, 3, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=2 的導數值。","answer":{"kind":"short_text","value":"124100","explanation_zh":"規則導數值=124100。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085099,"parameters":{"f":[11,4,2,1],"g":[18,1,3,1],"x":2,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"4\",\"2\",\"1\"],[\"18\",\"1\",\"3\",\"1\"],\"2\"],\"operation\":\"derivative_chain\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261008-r05-derivative-rules-0100","batch_id":"batch-20261007-advance-20261008-r05-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式合成的鏈式導數","stem_zh":"全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[12, 1, 0, 1]、g=[19, 1, 1, 1]，兩輸入多項式各至多三次；求 f∘g 在 x=-1 的導數值。","answer":{"kind":"short_text","value":"1946","explanation_zh":"規則導數值=1946。產生端用 f′(g(x))g′(x)，外層在內層值求導；合成可達九次，輸入次數上限不是中間式上限。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式合成的鏈式導數"]},"variant":{"family_id":"qf-derivative-rule-chain","seed":202610085100,"parameters":{"f":[12,1,0,1],"g":[19,1,1,1],"x":-1,"operation":"derivative_chain","domain":"reals","representation":"ascending_powers","semantic_template":"qf-derivative-rule-chain/{\"inputs\":[[\"1\",\"0\",\"1\"],[\"19\",\"1\",\"1\",\"1\"],\"-1\"],\"operation\":\"derivative_chain\"}"}},"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"}}