第 23050 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":[25,50,75,100],"order":4}。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":[25,50,75,100],"order":4}。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=[25, 50, 75, 100];declared order=4

declared degree BOUND=3=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

Higher derivative level1 ORIGINAL power1 coefficient product := mul(50, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 50

Higher derivative level1 ORIGINAL power2 coefficient product := mul(75, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 150

Higher derivative level1 ORIGINAL power3 coefficient product := mul(100, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 300

previous derivative order=0;actual low-to-high list=[25, 50, 75, 100];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 50, 1, 50], [2, 75, 2, 150], [3, 100, 3, 300]];next raw list=[50, 150, 300];all intermediate products SAFE

Higher derivative level2 ORIGINAL power1 coefficient product := mul(150, 1)

Higher derivative level2 ORIGINAL power1 coefficient product = 150

Higher derivative level2 ORIGINAL power2 coefficient product := mul(300, 2)

Higher derivative level2 ORIGINAL power2 coefficient product = 600

previous derivative order=1;actual low-to-high list=[50, 150, 300];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 150, 1, 150], [2, 300, 2, 600]];next raw list=[150, 600];all intermediate products SAFE

Higher derivative level3 ORIGINAL power1 coefficient product := mul(600, 1)

Higher derivative level3 ORIGINAL power1 coefficient product = 600

previous derivative order=2;actual low-to-high list=[150, 600];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 600, 1, 600]];next raw list=[600];all intermediate products SAFE

previous derivative order=3;actual low-to-high list=[600];drop ONLY its constant term

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

ORIGINAL coefficients=[25, 50, 75, 100];order=4;declared degree bound=3

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

• S013 · 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。
• S012 · 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。
• 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 level3 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 level2 ORIGINAL power2 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 level2 ORIGINAL power1 coefficient product，不以小數近似替代。
• S006 · 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。
• 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

(3, 4, 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。