{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r04-derivative-rules-0001","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[0]，求 f″(2)，採最簡正分母分數。","answer":{"kind":"expression","value":"0/1","explanation_zh":"二階導數=0/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094001,"parameters":{"f":[0],"x":2,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[\"zero_second_derivative\"]"}},"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-20261009-r04-derivative-rules-0002","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-11, 2, 2, 0]，求 f″(-1)，採最簡正分母分數。","answer":{"kind":"expression","value":"4/1","explanation_zh":"二階導數=4/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094002,"parameters":{"f":[-11,2,2,0],"x":-1,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[2],-1]"}},"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-20261009-r04-derivative-rules-0003","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-10, 3, 3, 1]，求 f″(0)，採最簡正分母分數。","answer":{"kind":"expression","value":"6/1","explanation_zh":"二階導數=6/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094003,"parameters":{"f":[-10,3,3,1],"x":0,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r04-derivative-rules-0004","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-9, 1, 4, -1]，求 f″(1)，採最簡正分母分數。","answer":{"kind":"expression","value":"2/1","explanation_zh":"二階導數=2/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094004,"parameters":{"f":[-9,1,4,-1],"x":1,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[4,-1],1]"}},"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-20261009-r04-derivative-rules-0005","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-8, 2, 5, 0]，求 f″(2)，採最簡正分母分數。","answer":{"kind":"expression","value":"10/1","explanation_zh":"二階導數=10/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094005,"parameters":{"f":[-8,2,5,0],"x":2,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[5],2]"}},"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-20261009-r04-derivative-rules-0006","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-7, 3, 6, 1]，求 f″(-2)，採最簡正分母分數。","answer":{"kind":"expression","value":"0/1","explanation_zh":"二階導數=0/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094006,"parameters":{"f":[-7,3,6,1],"x":-2,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[6,1],-2]"}},"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-20261009-r04-derivative-rules-0007","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-6, 1, 7, -1]，求 f″(-1)，採最簡正分母分數。","answer":{"kind":"expression","value":"20/1","explanation_zh":"二階導數=20/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094007,"parameters":{"f":[-6,1,7,-1],"x":-1,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[7,-1],-1]"}},"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-20261009-r04-derivative-rules-0008","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-5, 2, 8, 0]，求 f″(0)，採最簡正分母分數。","answer":{"kind":"expression","value":"16/1","explanation_zh":"二階導數=16/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094008,"parameters":{"f":[-5,2,8,0],"x":0,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[8]"}},"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-20261009-r04-derivative-rules-0009","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-4, 3, 9, 1]，求 f″(1)，採最簡正分母分數。","answer":{"kind":"expression","value":"24/1","explanation_zh":"二階導數=24/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094009,"parameters":{"f":[-4,3,9,1],"x":1,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[9,1],1]"}},"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-20261009-r04-derivative-rules-0010","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-3, 1, 10, -1]，求 f″(2)，採最簡正分母分數。","answer":{"kind":"expression","value":"8/1","explanation_zh":"二階導數=8/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094010,"parameters":{"f":[-3,1,10,-1],"x":2,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[10,-1],2]"}},"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-20261009-r04-derivative-rules-0011","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-2, 2, 11, 0]，求 f″(-2)，採最簡正分母分數。","answer":{"kind":"expression","value":"22/1","explanation_zh":"二階導數=22/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094011,"parameters":{"f":[-2,2,11,0],"x":-2,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[11],-2]"}},"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-20261009-r04-derivative-rules-0012","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-1, 3, 12, 1]，求 f″(-1)，採最簡正分母分數。","answer":{"kind":"expression","value":"18/1","explanation_zh":"二階導數=18/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094012,"parameters":{"f":[-1,3,12,1],"x":-1,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[12,1],-1]"}},"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-20261009-r04-derivative-rules-0013","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[0, 1, 13, -1]，求 f″(0)，採最簡正分母分數。","answer":{"kind":"expression","value":"26/1","explanation_zh":"二階導數=26/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094013,"parameters":{"f":[0,1,13,-1],"x":0,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[13]"}},"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-20261009-r04-derivative-rules-0014","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 2, 14, 0]，求 f″(1)，採最簡正分母分數。","answer":{"kind":"expression","value":"28/1","explanation_zh":"二階導數=28/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094014,"parameters":{"f":[1,2,14,0],"x":1,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[14],1]"}},"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-20261009-r04-derivative-rules-0015","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[2, 3, 15, 1]，求 f″(2)，採最簡正分母分數。","answer":{"kind":"expression","value":"42/1","explanation_zh":"二階導數=42/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094015,"parameters":{"f":[2,3,15,1],"x":2,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[15,1],2]"}},"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-20261009-r04-derivative-rules-0016","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[3, 1, 16, -1]，求 f″(-2)，採最簡正分母分數。","answer":{"kind":"expression","value":"44/1","explanation_zh":"二階導數=44/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094016,"parameters":{"f":[3,1,16,-1],"x":-2,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[16,-1],-2]"}},"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-20261009-r04-derivative-rules-0017","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[4, 2, 17, 0]，求 f″(-1)，採最簡正分母分數。","answer":{"kind":"expression","value":"34/1","explanation_zh":"二階導數=34/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094017,"parameters":{"f":[4,2,17,0],"x":-1,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[17],-1]"}},"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-20261009-r04-derivative-rules-0018","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[5, 3, 18, 1]，求 f″(0)，採最簡正分母分數。","answer":{"kind":"expression","value":"36/1","explanation_zh":"二階導數=36/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094018,"parameters":{"f":[5,3,18,1],"x":0,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[18]"}},"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-20261009-r04-derivative-rules-0019","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[6, 1, 19, -1]，求 f″(1)，採最簡正分母分數。","answer":{"kind":"expression","value":"32/1","explanation_zh":"二階導數=32/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094019,"parameters":{"f":[6,1,19,-1],"x":1,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[19,-1],1]"}},"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-20261009-r04-derivative-rules-0020","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[7, 2, 20, 0]，求 f″(2)，採最簡正分母分數。","answer":{"kind":"expression","value":"40/1","explanation_zh":"二階導數=40/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094020,"parameters":{"f":[7,2,20,0],"x":2,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[20],2]"}},"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-20261009-r04-derivative-rules-0021","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[8, 3, 21, 1]，求 f″(-2)，採最簡正分母分數。","answer":{"kind":"expression","value":"30/1","explanation_zh":"二階導數=30/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094021,"parameters":{"f":[8,3,21,1],"x":-2,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[21,1],-2]"}},"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-20261009-r04-derivative-rules-0022","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[9, 1, 22, -1]，求 f″(-1)，採最簡正分母分數。","answer":{"kind":"expression","value":"50/1","explanation_zh":"二階導數=50/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094022,"parameters":{"f":[9,1,22,-1],"x":-1,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[22,-1],-1]"}},"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-20261009-r04-derivative-rules-0023","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[10, 2, 23, 0]，求 f″(0)，採最簡正分母分數。","answer":{"kind":"expression","value":"46/1","explanation_zh":"二階導數=46/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094023,"parameters":{"f":[10,2,23,0],"x":0,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[23]"}},"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-20261009-r04-derivative-rules-0024","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[11, 3, 24, 1]，求 f″(1)，採最簡正分母分數。","answer":{"kind":"expression","value":"54/1","explanation_zh":"二階導數=54/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094024,"parameters":{"f":[11,3,24,1],"x":1,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[24,1],1]"}},"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-20261009-r04-derivative-rules-0025","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式的二階點導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[12, 1, 25, -1]，求 f″(2)，採最簡正分母分數。","answer":{"kind":"expression","value":"38/1","explanation_zh":"二階導數=38/1。產生端Horner同步傳遞值、一階及二階導數；常數及線性式二階導數為零，不以有限點抽樣當恆等證明。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式的二階點導數"]},"variant":{"family_id":"qf-density-derivative-second-value","seed":202610094025,"parameters":{"f":[12,1,25,-1],"x":2,"operation":"density_derivative_second_value","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-second-value/[[25,-1],2]"}},"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-20261009-r04-derivative-rules-0026","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[0]、g=[2, 1]，x=3，候選整數d=0。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=0/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094026,"parameters":{"f":[0],"g":[2,1],"x":3,"candidate":0,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[\"zero_derivative\",0]"}},"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-20261009-r04-derivative-rules-0027","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[5]、g=[-2]，x=1，候選整數d=1。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"否","explanation_zh":"乘積導數=0/1；候選資格=否。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094027,"parameters":{"f":[5],"g":[-2],"x":1,"candidate":1,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[\"zero_derivative\",1]"}},"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-20261009-r04-derivative-rules-0028","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-10, 3, 1, 3]、g=[2, -1, 1, 3]，x=1，候選整數d=40。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=40/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094028,"parameters":{"f":[-10,3,1,3],"g":[2,-1,1,3],"x":1,"candidate":40,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[-10,3,1,3],[2,-1,1,3]],1,40]"}},"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-20261009-r04-derivative-rules-0029","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-9, 4, -1, 4]、g=[2, -1, 1, 1]，x=2，候選整數d=981。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=981/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094029,"parameters":{"f":[-9,4,-1,4],"g":[2,-1,1,1],"x":2,"candidate":981,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[-9,4,-1,4],[2,-1,1,1]],2,981]"}},"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-20261009-r04-derivative-rules-0030","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-8, 5, 0, 1]、g=[2, -1, 1, 2]，x=-1，候選整數d=-25。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"否","explanation_zh":"乘積導數=-26/1；候選資格=否。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094030,"parameters":{"f":[-8,5,0,1],"g":[2,-1,1,2],"x":-1,"candidate":-25,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[-8,5,0,1],[2,-1,1,2]],-1,-25]"}},"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-20261009-r04-derivative-rules-0031","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-7, 6, 1, 2]、g=[2, -1, 1, 3]，x=0，候選整數d=19。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=19/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094031,"parameters":{"f":[-7,6,1,2],"g":[2,-1,1,3],"x":0,"candidate":19,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[-7,6,1,2],[2,-1,1,3]],0,19]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r04-derivative-rules-0032","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-6, 7, -1, 3]、g=[2, -1, 1, 1]，x=1，候選整數d=54。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=54/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094032,"parameters":{"f":[-6,7,-1,3],"g":[2,-1,1,1],"x":1,"candidate":54,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[-6,7,-1,3],[2,-1,1,1]],1,54]"}},"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-20261009-r04-derivative-rules-0033","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-5, 8, 0, 4]、g=[2, -1, 1, 2]，x=2，候選整數d=2282。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"否","explanation_zh":"乘積導數=2281/1；候選資格=否。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094033,"parameters":{"f":[-5,8,0,4],"g":[2,-1,1,2],"x":2,"candidate":2282,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[-5,8,0,4],[2,-1,1,2]],2,2282]"}},"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-20261009-r04-derivative-rules-0034","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-4, 9, 1, 1]、g=[2, -1, 1, 3]，x=-1，候選整數d=-68。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=-68/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094034,"parameters":{"f":[-4,9,1,1],"g":[2,-1,1,3],"x":-1,"candidate":-68,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[-4,9,1,1],[2,-1,1,3]],-1,-68]"}},"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-20261009-r04-derivative-rules-0035","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-3, 10, -1, 2]、g=[2, -1, 1, 1]，x=0，候選整數d=23。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=23/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094035,"parameters":{"f":[-3,10,-1,2],"g":[2,-1,1,1],"x":0,"candidate":23,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[-3,10,-1,2],[2,-1,1,1]],0,23]"}},"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-20261009-r04-derivative-rules-0036","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-2, 11, 0, 3]、g=[2, -1, 1, 2]，x=1，候選整數d=165。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"否","explanation_zh":"乘積導數=164/1；候選資格=否。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094036,"parameters":{"f":[-2,11,0,3],"g":[2,-1,1,2],"x":1,"candidate":165,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[-2,11,0,3],[2,-1,1,2]],1,165]"}},"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-20261009-r04-derivative-rules-0037","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-1, 12, 1, 4]、g=[2, -1, 1, 3]，x=2，候選整數d=4093。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=4093/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094037,"parameters":{"f":[-1,12,1,4],"g":[2,-1,1,3],"x":2,"candidate":4093,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[-1,12,1,4],[2,-1,1,3]],2,4093]"}},"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-20261009-r04-derivative-rules-0038","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[0, 13, -1, 1]、g=[2, -1, 1, 1]，x=-1，候選整數d=54。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=54/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094038,"parameters":{"f":[0,13,-1,1],"g":[2,-1,1,1],"x":-1,"candidate":54,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[0,13,-1,1],[2,-1,1,1]],-1,54]"}},"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-20261009-r04-derivative-rules-0039","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 14, 0, 2]、g=[2, -1, 1, 2]，x=0，候選整數d=28。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"否","explanation_zh":"乘積導數=27/1；候選資格=否。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094039,"parameters":{"f":[1,14,0,2],"g":[2,-1,1,2],"x":0,"candidate":28,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[1,14,0,2],[2,-1,1,2]],0,28]"}},"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-20261009-r04-derivative-rules-0040","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[2, 15, 1, 3]、g=[2, -1, 1, 3]，x=1，候選整數d=340。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=340/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094040,"parameters":{"f":[2,15,1,3],"g":[2,-1,1,3],"x":1,"candidate":340,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,3],[2,15,1,3]],1,340]"}},"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-20261009-r04-derivative-rules-0041","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[3, 16, -1, 4]、g=[2, -1, 1, 1]，x=2，候選整數d=1665。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=1665/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094041,"parameters":{"f":[3,16,-1,4],"g":[2,-1,1,1],"x":2,"candidate":1665,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,1],[3,16,-1,4]],2,1665]"}},"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-20261009-r04-derivative-rules-0042","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[4, 17, 0, 1]、g=[2, -1, 1, 2]，x=-1，候選整數d=-1。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"否","explanation_zh":"乘積導數=-2/1；候選資格=否。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094042,"parameters":{"f":[4,17,0,1],"g":[2,-1,1,2],"x":-1,"candidate":-1,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,2],[4,17,0,1]],-1,-1]"}},"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-20261009-r04-derivative-rules-0043","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[5, 18, 1, 2]、g=[2, -1, 1, 3]，x=0，候選整數d=31。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=31/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094043,"parameters":{"f":[5,18,1,2],"g":[2,-1,1,3],"x":0,"candidate":31,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,3],[5,18,1,2]],0,31]"}},"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-20261009-r04-derivative-rules-0044","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[6, 19, -1, 3]、g=[2, -1, 1, 1]，x=1，候選整數d=186。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=186/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094044,"parameters":{"f":[6,19,-1,3],"g":[2,-1,1,1],"x":1,"candidate":186,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,1],[6,19,-1,3]],1,186]"}},"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-20261009-r04-derivative-rules-0045","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[7, 20, 0, 4]、g=[2, -1, 1, 2]，x=2，候選整數d=3494。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"否","explanation_zh":"乘積導數=3493/1；候選資格=否。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094045,"parameters":{"f":[7,20,0,4],"g":[2,-1,1,2],"x":2,"candidate":3494,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,2],[7,20,0,4]],2,3494]"}},"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-20261009-r04-derivative-rules-0046","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[8, 21, 1, 1]、g=[2, -1, 1, 3]，x=-1，候選整數d=-56。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=-56/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094046,"parameters":{"f":[8,21,1,1],"g":[2,-1,1,3],"x":-1,"candidate":-56,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,3],[8,21,1,1]],-1,-56]"}},"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-20261009-r04-derivative-rules-0047","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[9, 22, -1, 2]、g=[2, -1, 1, 1]，x=0，候選整數d=35。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=35/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094047,"parameters":{"f":[9,22,-1,2],"g":[2,-1,1,1],"x":0,"candidate":35,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[2,-1,1,1],[9,22,-1,2]],0,35]"}},"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-20261009-r04-derivative-rules-0048","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[10, 23, 0, 3]、g=[2, -1, 1, 2]，x=1，候選整數d=381。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"否","explanation_zh":"乘積導數=380/1；候選資格=否。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094048,"parameters":{"f":[10,23,0,3],"g":[2,-1,1,2],"x":1,"candidate":381,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[10,23,0,3],[2,-1,1,2]],1,381]"}},"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-20261009-r04-derivative-rules-0049","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[11, 24, 1, 4]、g=[2, -1, 1, 3]，x=2，候選整數d=5833。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=5833/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094049,"parameters":{"f":[11,24,1,4],"g":[2,-1,1,3],"x":2,"candidate":5833,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[11,24,1,4],[2,-1,1,3]],2,5833]"}},"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-20261009-r04-derivative-rules-0050","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"乘積導數候選值的資格","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[12, 25, -1, 1]、g=[2, -1, 1, 1]，x=-1，候選整數d=90。d是否等於 (fg)′(x)？","answer":{"kind":"short_text","value":"是","explanation_zh":"乘積導數=90/1；候選資格=是。產生端用 f′g+fg′ 求精確導數後比較，不把兩導數乘積誤當此值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積導數候選值的資格"]},"variant":{"family_id":"qf-density-derivative-product-candidate","seed":202610094050,"parameters":{"f":[12,25,-1,1],"g":[2,-1,1,1],"x":-1,"candidate":90,"operation":"density_derivative_product_candidate","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-product-candidate/[[[12,25,-1,1],[2,-1,1,1]],-1,90]"}},"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-20261009-r04-derivative-rules-0051","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[3]、g=[1, 1]，x=2。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(0/1, 0/1)","explanation_zh":"有序合成導數=(0/1, 0/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094051,"parameters":{"f":[3],"g":[1,1],"x":2,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[\"both_constant_compositions\"]"}},"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-20261009-r04-derivative-rules-0052","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-11, 2, 0, 2]、g=[-4, 1, 2, 1]，x=0。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(98/1, 640/1)","explanation_zh":"有序合成導數=(98/1, 640/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094052,"parameters":{"f":[-11,2,0,2],"g":[-4,1,2,1],"x":0,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[-11,2,0,2],[-4,1,2,1],0]"}},"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-20261009-r04-derivative-rules-0053","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-10, 3, 1, 3]、g=[-3, 1, 3, 1]，x=1。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(430/1, 140/1)","explanation_zh":"有序合成導數=(430/1, 140/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094053,"parameters":{"f":[-10,3,1,3],"g":[-3,1,3,1],"x":1,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[-10,3,1,3],[-3,1,3,1],1]"}},"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-20261009-r04-derivative-rules-0054","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-9, 4, -1, 4]、g=[-2, 1, 1, 1]，x=2。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(29036/1, 107616/1)","explanation_zh":"有序合成導數=(29036/1, 107616/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094054,"parameters":{"f":[-9,4,-1,4],"g":[-2,1,1,1],"x":2,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[-9,4,-1,4],[-2,1,1,1],2]"}},"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-20261009-r04-derivative-rules-0055","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-8, 5, 0, 1]、g=[-1, 1, 2, 1]，x=-1。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(0/1, 4264/1)","explanation_zh":"有序合成導數=(0/1, 4264/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094055,"parameters":{"f":[-8,5,0,1],"g":[-1,1,2,1],"x":-1,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[-8,5,0,1],[-1,1,2,1],-1]"}},"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-20261009-r04-derivative-rules-0056","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-7, 6, 1, 2]、g=[0, 1, 3, 1]，x=0。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(6/1, 636/1)","explanation_zh":"有序合成導數=(6/1, 636/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094056,"parameters":{"f":[-7,6,1,2],"g":[0,1,3,1],"x":0,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[-7,6,1,2],[0,1,3,1],0]"}},"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-20261009-r04-derivative-rules-0057","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-6, 7, -1, 3]、g=[1, 1, 1, 1]，x=1。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(858/1, 476/1)","explanation_zh":"有序合成導數=(858/1, 476/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094057,"parameters":{"f":[-6,7,-1,3],"g":[1,1,1,1],"x":1,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[-6,7,-1,3],[1,1,1,1],1]"}},"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-20261009-r04-derivative-rules-0058","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-5, 8, 0, 4]、g=[2, 1, 2, 1]，x=2。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(100968/1, 320320/1)","explanation_zh":"有序合成導數=(100968/1, 320320/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094058,"parameters":{"f":[-5,8,0,4],"g":[2,1,2,1],"x":2,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[-5,8,0,4],[2,1,2,1],2]"}},"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-20261009-r04-derivative-rules-0059","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-4, 9, 1, 1]、g=[3, 1, 3, 1]，x=-1。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(-130/1, 4300/1)","explanation_zh":"有序合成導數=(-130/1, 4300/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094059,"parameters":{"f":[-4,9,1,1],"g":[3,1,3,1],"x":-1,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[-4,9,1,1],[3,1,3,1],-1]"}},"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-20261009-r04-derivative-rules-0060","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-3, 10, -1, 2]、g=[4, 1, 1, 1]，x=0。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(98/1, 220/1)","explanation_zh":"有序合成導數=(98/1, 220/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094060,"parameters":{"f":[-3,10,-1,2],"g":[4,1,1,1],"x":0,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[-3,10,-1,2],[4,1,1,1],0]"}},"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-20261009-r04-derivative-rules-0061","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-2, 11, 0, 3]、g=[5, 1, 2, 1]，x=1。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(5920/1, 9620/1)","explanation_zh":"有序合成導數=(5920/1, 9620/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094061,"parameters":{"f":[-2,11,0,3],"g":[5,1,2,1],"x":1,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[-2,11,0,3],[5,1,2,1],1]"}},"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-20261009-r04-derivative-rules-0062","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[-1, 12, 1, 4]、g=[6, 1, 3, 1]，x=2。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(236900/1, 691072/1)","explanation_zh":"有序合成導數=(236900/1, 691072/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094062,"parameters":{"f":[-1,12,1,4],"g":[6,1,3,1],"x":2,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[-1,12,1,4],[6,1,3,1],2]"}},"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-20261009-r04-derivative-rules-0063","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[0, 13, -1, 1]、g=[7, 1, 1, 1]，x=-1。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(218/1, 11628/1)","explanation_zh":"有序合成導數=(218/1, 11628/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094063,"parameters":{"f":[0,13,-1,1],"g":[7,1,1,1],"x":-1,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[0,13,-1,1],[7,1,1,1],-1]"}},"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-20261009-r04-derivative-rules-0064","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 14, 0, 2]、g=[8, 1, 2, 1]，x=0。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(398/1, 112/1)","explanation_zh":"有序合成導數=(398/1, 112/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094064,"parameters":{"f":[1,14,0,2],"g":[8,1,2,1],"x":0,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[1,14,0,2],[8,1,2,1],0]"}},"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-20261009-r04-derivative-rules-0065","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[2, 15, 1, 3]、g=[9, 1, 3, 1]，x=1。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(18070/1, 37700/1)","explanation_zh":"有序合成導數=(18070/1, 37700/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094065,"parameters":{"f":[2,15,1,3],"g":[9,1,3,1],"x":1,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[2,15,1,3],[9,1,3,1],1]"}},"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-20261009-r04-derivative-rules-0066","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[3, 16, -1, 4]、g=[10, 1, 1, 1]，x=2。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(116960/1, 722040/1)","explanation_zh":"有序合成導數=(116960/1, 722040/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094066,"parameters":{"f":[3,16,-1,4],"g":[10,1,1,1],"x":2,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[3,16,-1,4],[10,1,1,1],2]"}},"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-20261009-r04-derivative-rules-0067","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[4, 17, 0, 1]、g=[11, 1, 2, 1]，x=-1。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(0/1, 10660/1)","explanation_zh":"有序合成導數=(0/1, 10660/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094067,"parameters":{"f":[4,17,0,1],"g":[11,1,2,1],"x":-1,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[4,17,0,1],[11,1,2,1],-1]"}},"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-20261009-r04-derivative-rules-0068","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[5, 18, 1, 2]、g=[12, 1, 3, 1]，x=0。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(906/1, 1908/1)","explanation_zh":"有序合成導數=(906/1, 1908/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094068,"parameters":{"f":[5,18,1,2],"g":[12,1,3,1],"x":0,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[5,18,1,2],[12,1,3,1],0]"}},"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-20261009-r04-derivative-rules-0069","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[6, 19, -1, 3]、g=[13, 1, 1, 1]，x=1。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(13746/1, 58292/1)","explanation_zh":"有序合成導數=(13746/1, 58292/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094069,"parameters":{"f":[6,19,-1,3],"g":[13,1,1,1],"x":1,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[6,19,-1,3],[13,1,1,1],1]"}},"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-20261009-r04-derivative-rules-0070","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[7, 20, 0, 4]、g=[14, 1, 2, 1]，x=2。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(258468/1, 1294720/1)","explanation_zh":"有序合成導數=(258468/1, 1294720/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094070,"parameters":{"f":[7,20,0,4],"g":[14,1,2,1],"x":2,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[7,20,0,4],[14,1,2,1],2]"}},"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-20261009-r04-derivative-rules-0071","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[8, 21, 1, 1]、g=[15, 1, 3, 1]，x=-1。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(-1642/1, 9460/1)","explanation_zh":"有序合成導數=(-1642/1, 9460/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094071,"parameters":{"f":[8,21,1,1],"g":[15,1,3,1],"x":-1,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[8,21,1,1],[15,1,3,1],-1]"}},"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-20261009-r04-derivative-rules-0072","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[9, 22, -1, 2]、g=[16, 1, 1, 1]，x=0。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(1526/1, 5764/1)","explanation_zh":"有序合成導數=(1526/1, 5764/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094072,"parameters":{"f":[9,22,-1,2],"g":[16,1,1,1],"x":0,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[9,22,-1,2],[16,1,1,1],0]"}},"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-20261009-r04-derivative-rules-0073","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[10, 23, 0, 3]、g=[17, 1, 2, 1]，x=1。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(31936/1, 129056/1)","explanation_zh":"有序合成導數=(31936/1, 129056/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094073,"parameters":{"f":[10,23,0,3],"g":[17,1,2,1],"x":1,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[10,23,0,3],[17,1,2,1],1]"}},"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-20261009-r04-derivative-rules-0074","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[11, 24, 1, 4]、g=[18, 1, 3, 1]，x=2。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(482600/1, 2101096/1)","explanation_zh":"有序合成導數=(482600/1, 2101096/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094074,"parameters":{"f":[11,24,1,4],"g":[18,1,3,1],"x":2,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[11,24,1,4],[18,1,3,1],2]"}},"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-20261009-r04-derivative-rules-0075","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"兩個合成次序的導數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[12, 25, -1, 1]、g=[19, 1, 1, 1]，x=-1。依序列 ((f∘g)′(x), (g∘f)′(x))，每項保留 /1 或最簡分數。","answer":{"kind":"expression","value":"(1922/1, 19380/1)","explanation_zh":"有序合成導數=(1922/1, 19380/1)。外函數在內函數值取導數，再乘內導數；交換次序可能改變值，常數合成則為零。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["兩個合成次序的導數"]},"variant":{"family_id":"qf-density-derivative-chain-order-pair","seed":202610094075,"parameters":{"f":[12,25,-1,1],"g":[19,1,1,1],"x":-1,"operation":"density_derivative_chain_order_pair","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-chain-order-pair/[[12,25,-1,1],[19,1,1,1],-1]"}},"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-20261009-r04-derivative-rules-0076","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[2, 1]、g=[-4, -1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[0]","explanation_zh":"和的導數升冪係數=[0]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094076,"parameters":{"f":[2,1],"g":[-4,-1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,-1],[0,1]]"}},"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-20261009-r04-derivative-rules-0077","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 1, 3, 2]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[2, 8, 9]","explanation_zh":"和的導數升冪係數=[2, 8, 9]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094077,"parameters":{"f":[1,1,3,2],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,1,3,2]]"}},"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-20261009-r04-derivative-rules-0078","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 2, 4, 3]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[3, 10, 12]","explanation_zh":"和的導數升冪係數=[3, 10, 12]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094078,"parameters":{"f":[1,2,4,3],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,2,4,3]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r04-derivative-rules-0079","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 3, 5, 1]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[4, 12, 6]","explanation_zh":"和的導數升冪係數=[4, 12, 6]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094079,"parameters":{"f":[1,3,5,1],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,3,5,1]]"}},"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-20261009-r04-derivative-rules-0080","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 4, 6, 2]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[5, 14, 9]","explanation_zh":"和的導數升冪係數=[5, 14, 9]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094080,"parameters":{"f":[1,4,6,2],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,4,6,2]]"}},"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-20261009-r04-derivative-rules-0081","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 5, 7, 3]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[6, 16, 12]","explanation_zh":"和的導數升冪係數=[6, 16, 12]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094081,"parameters":{"f":[1,5,7,3],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,5,7,3]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r04-derivative-rules-0082","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 6, 8, 1]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[7, 18, 6]","explanation_zh":"和的導數升冪係數=[7, 18, 6]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094082,"parameters":{"f":[1,6,8,1],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,6,8,1]]"}},"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-20261009-r04-derivative-rules-0083","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 7, 9, 2]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[8, 20, 9]","explanation_zh":"和的導數升冪係數=[8, 20, 9]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094083,"parameters":{"f":[1,7,9,2],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,7,9,2]]"}},"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-20261009-r04-derivative-rules-0084","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 8, 10, 3]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[9, 22, 12]","explanation_zh":"和的導數升冪係數=[9, 22, 12]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094084,"parameters":{"f":[1,8,10,3],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,8,10,3]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r04-derivative-rules-0085","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 9, 11, 1]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[10, 24, 6]","explanation_zh":"和的導數升冪係數=[10, 24, 6]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094085,"parameters":{"f":[1,9,11,1],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,9,11,1]]"}},"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-20261009-r04-derivative-rules-0086","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 10, 12, 2]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[11, 26, 9]","explanation_zh":"和的導數升冪係數=[11, 26, 9]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094086,"parameters":{"f":[1,10,12,2],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,10,12,2]]"}},"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-20261009-r04-derivative-rules-0087","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 11, 13, 3]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[12, 28, 12]","explanation_zh":"和的導數升冪係數=[12, 28, 12]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094087,"parameters":{"f":[1,11,13,3],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,11,13,3]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r04-derivative-rules-0088","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 12, 14, 1]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[13, 30, 6]","explanation_zh":"和的導數升冪係數=[13, 30, 6]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094088,"parameters":{"f":[1,12,14,1],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,12,14,1]]"}},"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-20261009-r04-derivative-rules-0089","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 13, 15, 2]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[14, 32, 9]","explanation_zh":"和的導數升冪係數=[14, 32, 9]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094089,"parameters":{"f":[1,13,15,2],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,13,15,2]]"}},"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-20261009-r04-derivative-rules-0090","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 14, 16, 3]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[15, 34, 12]","explanation_zh":"和的導數升冪係數=[15, 34, 12]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094090,"parameters":{"f":[1,14,16,3],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,14,16,3]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r04-derivative-rules-0091","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 15, 17, 1]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[16, 36, 6]","explanation_zh":"和的導數升冪係數=[16, 36, 6]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094091,"parameters":{"f":[1,15,17,1],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,15,17,1]]"}},"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-20261009-r04-derivative-rules-0092","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 16, 18, 2]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[17, 38, 9]","explanation_zh":"和的導數升冪係數=[17, 38, 9]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094092,"parameters":{"f":[1,16,18,2],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,16,18,2]]"}},"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-20261009-r04-derivative-rules-0093","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 17, 19, 3]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[18, 40, 12]","explanation_zh":"和的導數升冪係數=[18, 40, 12]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094093,"parameters":{"f":[1,17,19,3],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,17,19,3]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r04-derivative-rules-0094","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 18, 20, 1]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[19, 42, 6]","explanation_zh":"和的導數升冪係數=[19, 42, 6]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094094,"parameters":{"f":[1,18,20,1],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,18,20,1]]"}},"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-20261009-r04-derivative-rules-0095","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 19, 21, 2]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[20, 44, 9]","explanation_zh":"和的導數升冪係數=[20, 44, 9]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094095,"parameters":{"f":[1,19,21,2],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,19,21,2]]"}},"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-20261009-r04-derivative-rules-0096","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 20, 22, 3]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[21, 46, 12]","explanation_zh":"和的導數升冪係數=[21, 46, 12]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094096,"parameters":{"f":[1,20,22,3],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,20,22,3]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r04-derivative-rules-0097","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 21, 23, 1]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[22, 48, 6]","explanation_zh":"和的導數升冪係數=[22, 48, 6]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094097,"parameters":{"f":[1,21,23,1],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,21,23,1]]"}},"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-20261009-r04-derivative-rules-0098","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 22, 24, 2]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[23, 50, 9]","explanation_zh":"和的導數升冪係數=[23, 50, 9]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094098,"parameters":{"f":[1,22,24,2],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,22,24,2]]"}},"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-20261009-r04-derivative-rules-0099","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 23, 25, 3]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[24, 52, 12]","explanation_zh":"和的導數升冪係數=[24, 52, 12]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094099,"parameters":{"f":[1,23,25,3],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,23,25,3]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-07","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261007-advance-20261009-r04-derivative-rules-0100","batch_id":"batch-20261007-advance-20261009-r04-derivative-rules-foundations","language":"zh-Hant","source_mko_ids":["mko-derivative-rules"],"title_zh":"多項式和的完整導數係數","stem_zh":"全實數論域；升冪係數的受限多項式模型。升冪係數 f=[1, 24, 26, 1]、g=[-1, 1, 1, 1]。列 (f+g)′ 的完整升冪整數係數，末端零移除、零式列[0]。","answer":{"kind":"expression","value":"[25, 54, 6]","explanation_zh":"和的導數升冪係數=[25, 54, 6]。逐次項加權形成係數恆等，常數項不參與；沒有以單點值代替完整列表。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["多項式和的完整導數係數"]},"variant":{"family_id":"qf-density-derivative-sum-coefficients","seed":202610094100,"parameters":{"f":[1,24,26,1],"g":[-1,1,1,1],"operation":"density_derivative_sum_coefficients","domain":"reals","representation":"ascending_powers","semantic_template":"qf-density-derivative-sum-coefficients/[[0,1,1,1],[0,24,26,1]]"}},"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"}}