第 23057 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":[7,14,21,28,35,42],"order":6,"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":[7,14,21,28,35,42],"order":6,"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=[7, 14, 21, 28, 35, 42];declared order=6

declared degree BOUND=5=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

Higher derivative level1 ORIGINAL power1 coefficient product := mul(14, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 14

Higher derivative level1 ORIGINAL power2 coefficient product := mul(21, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 42

Higher derivative level1 ORIGINAL power3 coefficient product := mul(28, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 84

Higher derivative level1 ORIGINAL power4 coefficient product := mul(35, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 140

Higher derivative level1 ORIGINAL power5 coefficient product := mul(42, 5)

Higher derivative level1 ORIGINAL power5 coefficient product = 210

previous derivative order=0;actual low-to-high list=[7, 14, 21, 28, 35, 42];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 14, 1, 14], [2, 21, 2, 42], [3, 28, 3, 84], [4, 35, 4, 140], [5, 42, 5, 210]];next raw list=[14, 42, 84, 140, 210];all intermediate products SAFE

Higher derivative level2 ORIGINAL power1 coefficient product := mul(42, 1)

Higher derivative level2 ORIGINAL power1 coefficient product = 42

Higher derivative level2 ORIGINAL power2 coefficient product := mul(84, 2)

Higher derivative level2 ORIGINAL power2 coefficient product = 168

Higher derivative level2 ORIGINAL power3 coefficient product := mul(140, 3)

Higher derivative level2 ORIGINAL power3 coefficient product = 420

Higher derivative level2 ORIGINAL power4 coefficient product := mul(210, 4)

Higher derivative level2 ORIGINAL power4 coefficient product = 840

previous derivative order=1;actual low-to-high list=[14, 42, 84, 140, 210];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 42, 1, 42], [2, 84, 2, 168], [3, 140, 3, 420], [4, 210, 4, 840]];next raw list=[42, 168, 420, 840];all intermediate products SAFE

Higher derivative level3 ORIGINAL power1 coefficient product := mul(168, 1)

Higher derivative level3 ORIGINAL power1 coefficient product = 168

Higher derivative level3 ORIGINAL power2 coefficient product := mul(420, 2)

Higher derivative level3 ORIGINAL power2 coefficient product = 840

Higher derivative level3 ORIGINAL power3 coefficient product := mul(840, 3)

Higher derivative level3 ORIGINAL power3 coefficient product = 2520

previous derivative order=2;actual low-to-high list=[42, 168, 420, 840];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 168, 1, 168], [2, 420, 2, 840], [3, 840, 3, 2520]];next raw list=[168, 840, 2520];all intermediate products SAFE

Higher derivative level4 ORIGINAL power1 coefficient product := mul(840, 1)

Higher derivative level4 ORIGINAL power1 coefficient product = 840

Higher derivative level4 ORIGINAL power2 coefficient product := mul(2520, 2)

Higher derivative level4 ORIGINAL power2 coefficient product = 5040

previous derivative order=3;actual low-to-high list=[168, 840, 2520];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 840, 1, 840], [2, 2520, 2, 5040]];next raw list=[840, 5040];all intermediate products SAFE

Higher derivative level5 ORIGINAL power1 coefficient product := mul(5040, 1)

Higher derivative level5 ORIGINAL power1 coefficient product = 5040

previous derivative order=4;actual low-to-high list=[840, 5040];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 5040, 1, 5040]];next raw list=[5040];all intermediate products SAFE

previous derivative order=5;actual low-to-high list=[5040];drop ONLY its constant term

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

ORIGINAL coefficients=[7, 14, 21, 28, 35, 42];order=6;declared degree bound=5

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

Higher derivative Horner power0 previous value times ORIGINAL point := mul(0, -1)

Higher derivative Horner power0 previous value times ORIGINAL point = 0

Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(0, 0)

Higher derivative Horner power0 plus ORIGINAL derivative coefficient = 0

逐步反向核對

• S026 · 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。
• S025 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S022 · 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。
• S021 · 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。
• 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 level4 ORIGINAL power2 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 level4 ORIGINAL power1 coefficient product，不以小數近似替代。
• 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 — 以精確有理數重算 Higher derivative level3 ORIGINAL power3 coefficient product，不以小數近似替代。
• S015 · 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。
• S014 · 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。
• 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 — 以精確有理數重算 Higher derivative level2 ORIGINAL power4 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 power3 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 power2 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 power1 coefficient product，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 — 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

([7, 14, 21, 28, 35, 42], 6, [0], -1, 0, 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。