第 23031 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":[6,12,18,24,30],"order":5}。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":[6,12,18,24,30],"order":5}。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=[6, 12, 18, 24, 30];declared order=5

declared degree BOUND=4=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

Higher derivative level1 ORIGINAL power1 coefficient product := mul(12, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 12

Higher derivative level1 ORIGINAL power2 coefficient product := mul(18, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 36

Higher derivative level1 ORIGINAL power3 coefficient product := mul(24, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 72

Higher derivative level1 ORIGINAL power4 coefficient product := mul(30, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 120

previous derivative order=0;actual low-to-high list=[6, 12, 18, 24, 30];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 12, 1, 12], [2, 18, 2, 36], [3, 24, 3, 72], [4, 30, 4, 120]];next raw list=[12, 36, 72, 120];all intermediate products SAFE

Higher derivative level2 ORIGINAL power1 coefficient product := mul(36, 1)

Higher derivative level2 ORIGINAL power1 coefficient product = 36

Higher derivative level2 ORIGINAL power2 coefficient product := mul(72, 2)

Higher derivative level2 ORIGINAL power2 coefficient product = 144

Higher derivative level2 ORIGINAL power3 coefficient product := mul(120, 3)

Higher derivative level2 ORIGINAL power3 coefficient product = 360

previous derivative order=1;actual low-to-high list=[12, 36, 72, 120];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 36, 1, 36], [2, 72, 2, 144], [3, 120, 3, 360]];next raw list=[36, 144, 360];all intermediate products SAFE

Higher derivative level3 ORIGINAL power1 coefficient product := mul(144, 1)

Higher derivative level3 ORIGINAL power1 coefficient product = 144

Higher derivative level3 ORIGINAL power2 coefficient product := mul(360, 2)

Higher derivative level3 ORIGINAL power2 coefficient product = 720

previous derivative order=2;actual low-to-high list=[36, 144, 360];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 144, 1, 144], [2, 360, 2, 720]];next raw list=[144, 720];all intermediate products SAFE

Higher derivative level4 ORIGINAL power1 coefficient product := mul(720, 1)

Higher derivative level4 ORIGINAL power1 coefficient product = 720

previous derivative order=3;actual low-to-high list=[144, 720];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 720, 1, 720]];next raw list=[720];all intermediate products SAFE

previous derivative order=4;actual low-to-high list=[720];drop ONLY its constant term

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

ORIGINAL coefficients=[6, 12, 18, 24, 30];order=5;declared degree bound=4

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

• S018 · 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。
• S017 · 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。
• S016 · 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。
• S015 · 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。
• S014 · 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。
• S013 · 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。
• S012 · 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。
• S011 · 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。
• S010 · 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。
• S009 · 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。
• S008 · 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。
• S007 · 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。
• 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

(4, 5, 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。