第 23041 題：高階導數

原題與已知條件

f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={"coefficients":[16,32,48,64,80,96,112],"order":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — f^(0)=f且f^(n+1)=(f^n)'的逐階存在與n in natural including zero。操作=higher-order-qualification；資料={"coefficients":[16,32,48,64,80,96,112],"order":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-higher-order-derivative

逐步推導

source operation=higher-order-qualification;original snapshot and mathematics retained

explicit ordered source output fields=[declared_degree_bound, order, nonnegative_order, order_convention, polynomial_chain_certificate]

ORIGINAL low-to-high coefficients=[16, 32, 48, 64, 80, 96, 112];declared order=7

declared degree BOUND=6=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

Higher derivative level1 ORIGINAL power1 coefficient product := mul(32, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 32

Higher derivative level1 ORIGINAL power2 coefficient product := mul(48, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 96

Higher derivative level1 ORIGINAL power3 coefficient product := mul(64, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 192

Higher derivative level1 ORIGINAL power4 coefficient product := mul(80, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 320

Higher derivative level1 ORIGINAL power5 coefficient product := mul(96, 5)

Higher derivative level1 ORIGINAL power5 coefficient product = 480

Higher derivative level1 ORIGINAL power6 coefficient product := mul(112, 6)

Higher derivative level1 ORIGINAL power6 coefficient product = 672

previous derivative order=0;actual low-to-high list=[16, 32, 48, 64, 80, 96, 112];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 32, 1, 32], [2, 48, 2, 96], [3, 64, 3, 192], [4, 80, 4, 320], [5, 96, 5, 480], [6, 112, 6, 672]];next raw list=[32, 96, 192, 320, 480, 672];all intermediate products SAFE

Higher derivative level2 ORIGINAL power1 coefficient product := mul(96, 1)

Higher derivative level2 ORIGINAL power1 coefficient product = 96

Higher derivative level2 ORIGINAL power2 coefficient product := mul(192, 2)

Higher derivative level2 ORIGINAL power2 coefficient product = 384

Higher derivative level2 ORIGINAL power3 coefficient product := mul(320, 3)

Higher derivative level2 ORIGINAL power3 coefficient product = 960

Higher derivative level2 ORIGINAL power4 coefficient product := mul(480, 4)

Higher derivative level2 ORIGINAL power4 coefficient product = 1920

Higher derivative level2 ORIGINAL power5 coefficient product := mul(672, 5)

Higher derivative level2 ORIGINAL power5 coefficient product = 3360

previous derivative order=1;actual low-to-high list=[32, 96, 192, 320, 480, 672];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 96, 1, 96], [2, 192, 2, 384], [3, 320, 3, 960], [4, 480, 4, 1920], [5, 672, 5, 3360]];next raw list=[96, 384, 960, 1920, 3360];all intermediate products SAFE

Higher derivative level3 ORIGINAL power1 coefficient product := mul(384, 1)

Higher derivative level3 ORIGINAL power1 coefficient product = 384

Higher derivative level3 ORIGINAL power2 coefficient product := mul(960, 2)

Higher derivative level3 ORIGINAL power2 coefficient product = 1920

Higher derivative level3 ORIGINAL power3 coefficient product := mul(1920, 3)

Higher derivative level3 ORIGINAL power3 coefficient product = 5760

Higher derivative level3 ORIGINAL power4 coefficient product := mul(3360, 4)

Higher derivative level3 ORIGINAL power4 coefficient product = 13440

previous derivative order=2;actual low-to-high list=[96, 384, 960, 1920, 3360];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 384, 1, 384], [2, 960, 2, 1920], [3, 1920, 3, 5760], [4, 3360, 4, 13440]];next raw list=[384, 1920, 5760, 13440];all intermediate products SAFE

Higher derivative level4 ORIGINAL power1 coefficient product := mul(1920, 1)

Higher derivative level4 ORIGINAL power1 coefficient product = 1920

Higher derivative level4 ORIGINAL power2 coefficient product := mul(5760, 2)

Higher derivative level4 ORIGINAL power2 coefficient product = 11520

Higher derivative level4 ORIGINAL power3 coefficient product := mul(13440, 3)

Higher derivative level4 ORIGINAL power3 coefficient product = 40320

previous derivative order=3;actual low-to-high list=[384, 1920, 5760, 13440];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 1920, 1, 1920], [2, 5760, 2, 11520], [3, 13440, 3, 40320]];next raw list=[1920, 11520, 40320];all intermediate products SAFE

Higher derivative level5 ORIGINAL power1 coefficient product := mul(11520, 1)

Higher derivative level5 ORIGINAL power1 coefficient product = 11520

Higher derivative level5 ORIGINAL power2 coefficient product := mul(40320, 2)

Higher derivative level5 ORIGINAL power2 coefficient product = 80640

previous derivative order=4;actual low-to-high list=[1920, 11520, 40320];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 11520, 1, 11520], [2, 40320, 2, 80640]];next raw list=[11520, 80640];all intermediate products SAFE

Higher derivative level6 ORIGINAL power1 coefficient product := mul(80640, 1)

Higher derivative level6 ORIGINAL power1 coefficient product = 80640

previous derivative order=5;actual low-to-high list=[11520, 80640];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 80640, 1, 80640]];next raw list=[80640];all intermediate products SAFE

previous derivative order=6;actual low-to-high list=[80640];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[];next raw list=[];all intermediate products SAFE

ORIGINAL coefficients=[16, 32, 48, 64, 80, 96, 112];order=7;declared degree bound=6

final derivative coefficients=[0];for each original power k>=order multiplier=k*(k-1)*...*(k-order+1);lower powers vanish

逐步反向核對

• S031 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S030 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S029 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S028 · premise_and_result_audit — 以精確有理數重算 Higher derivative level6 ORIGINAL power1 coefficient product，不以小數近似替代。
• S027 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S026 · premise_and_result_audit — 以精確有理數重算 Higher derivative level5 ORIGINAL power2 coefficient product，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Higher derivative level5 ORIGINAL power1 coefficient product，不以小數近似替代。
• S024 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S023 · premise_and_result_audit — 以精確有理數重算 Higher derivative level4 ORIGINAL power3 coefficient product，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Higher derivative level4 ORIGINAL power2 coefficient product，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Higher derivative level4 ORIGINAL power1 coefficient product，不以小數近似替代。
• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · premise_and_result_audit — 以精確有理數重算 Higher derivative level3 ORIGINAL power4 coefficient product，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Higher derivative level3 ORIGINAL power3 coefficient product，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Higher derivative level3 ORIGINAL power2 coefficient product，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Higher derivative level3 ORIGINAL power1 coefficient product，不以小數近似替代。
• S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S014 · premise_and_result_audit — 以精確有理數重算 Higher derivative level2 ORIGINAL power5 coefficient product，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Higher derivative level2 ORIGINAL power4 coefficient product，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Higher derivative level2 ORIGINAL power3 coefficient product，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Higher derivative level2 ORIGINAL power2 coefficient product，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Higher derivative level2 ORIGINAL power1 coefficient product，不以小數近似替代。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power6 coefficient product，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power5 coefficient product，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power4 coefficient product，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power3 coefficient product，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power2 coefficient product，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Higher derivative level1 ORIGINAL power1 coefficient product，不以小數近似替代。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(6, 7, 是, recursive_derivative_chain_required, 是)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。