第 23064 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-polynomial-nth-derivative；資料={"coefficients":[14,28,42,56,70],"order":1,"point":1}。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-polynomial-nth-derivative；資料={"coefficients":[14,28,42,56,70],"order":1,"point":1}。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-polynomial-nth-derivative;original snapshot and mathematics retained

explicit ordered source output fields=[coefficients, order, derivative_coefficients, point, value, certificate]

ORIGINAL low-to-high coefficients=[14, 28, 42, 56, 70];declared order=1

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

Higher derivative level1 ORIGINAL power1 coefficient product := mul(28, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 28

Higher derivative level1 ORIGINAL power2 coefficient product := mul(42, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 84

Higher derivative level1 ORIGINAL power3 coefficient product := mul(56, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 168

Higher derivative level1 ORIGINAL power4 coefficient product := mul(70, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 280

previous derivative order=0;actual low-to-high list=[14, 28, 42, 56, 70];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 28, 1, 28], [2, 42, 2, 84], [3, 56, 3, 168], [4, 70, 4, 280]];next raw list=[28, 84, 168, 280];all intermediate products SAFE

ORIGINAL coefficients=[14, 28, 42, 56, 70];order=1;declared degree bound=4

final derivative coefficients=[28, 84, 168, 280];for each original power k>=order multiplier=k*(k-1)*...*(k-order+1);lower powers vanish

Higher derivative Horner power3 previous value times ORIGINAL point := mul(0, 1)

Higher derivative Horner power3 previous value times ORIGINAL point = 0

Higher derivative Horner power3 plus ORIGINAL derivative coefficient := sum(0, 280)

Higher derivative Horner power3 plus ORIGINAL derivative coefficient = 280

Higher derivative Horner power2 previous value times ORIGINAL point := mul(280, 1)

Higher derivative Horner power2 previous value times ORIGINAL point = 280

Higher derivative Horner power2 plus ORIGINAL derivative coefficient := sum(280, 168)

Higher derivative Horner power2 plus ORIGINAL derivative coefficient = 448

Higher derivative Horner power1 previous value times ORIGINAL point := mul(448, 1)

Higher derivative Horner power1 previous value times ORIGINAL point = 448

Higher derivative Horner power1 plus ORIGINAL derivative coefficient := sum(448, 84)

Higher derivative Horner power1 plus ORIGINAL derivative coefficient = 532

Higher derivative Horner power0 previous value times ORIGINAL point := mul(532, 1)

Higher derivative Horner power0 previous value times ORIGINAL point = 532

Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(532, 28)

Higher derivative Horner power0 plus ORIGINAL derivative coefficient = 560

逐步反向核對

• S016 · premise_and_result_audit — 以精確有理數重算 Higher derivative Horner power0 plus ORIGINAL derivative coefficient，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Higher derivative Horner power0 previous value times ORIGINAL point，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Higher derivative Horner power1 plus ORIGINAL derivative coefficient，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Higher derivative Horner power1 previous value times ORIGINAL point，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Higher derivative Horner power2 plus ORIGINAL derivative coefficient，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Higher derivative Horner power2 previous value times ORIGINAL point，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Higher derivative Horner power3 plus ORIGINAL derivative coefficient，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Higher derivative Horner power3 previous value times ORIGINAL point，不以小數近似替代。
• S008 · 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。
• 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

([14, 28, 42, 56, 70], 1, [28, 84, 168, 280], 1, 560, falling_factorial_coefficients_exact)

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。