第 23061 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":[11,22,33,44,55,66],"order":3,"point":-2}。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":[11,22,33,44,55,66],"order":3,"point":-2}。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=[11, 22, 33, 44, 55, 66];declared order=3

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

Higher derivative level1 ORIGINAL power1 coefficient product := mul(22, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 22

Higher derivative level1 ORIGINAL power2 coefficient product := mul(33, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 66

Higher derivative level1 ORIGINAL power3 coefficient product := mul(44, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 132

Higher derivative level1 ORIGINAL power4 coefficient product := mul(55, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 220

Higher derivative level1 ORIGINAL power5 coefficient product := mul(66, 5)

Higher derivative level1 ORIGINAL power5 coefficient product = 330

previous derivative order=0;actual low-to-high list=[11, 22, 33, 44, 55, 66];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 22, 1, 22], [2, 33, 2, 66], [3, 44, 3, 132], [4, 55, 4, 220], [5, 66, 5, 330]];next raw list=[22, 66, 132, 220, 330];all intermediate products SAFE

Higher derivative level2 ORIGINAL power1 coefficient product := mul(66, 1)

Higher derivative level2 ORIGINAL power1 coefficient product = 66

Higher derivative level2 ORIGINAL power2 coefficient product := mul(132, 2)

Higher derivative level2 ORIGINAL power2 coefficient product = 264

Higher derivative level2 ORIGINAL power3 coefficient product := mul(220, 3)

Higher derivative level2 ORIGINAL power3 coefficient product = 660

Higher derivative level2 ORIGINAL power4 coefficient product := mul(330, 4)

Higher derivative level2 ORIGINAL power4 coefficient product = 1320

previous derivative order=1;actual low-to-high list=[22, 66, 132, 220, 330];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 66, 1, 66], [2, 132, 2, 264], [3, 220, 3, 660], [4, 330, 4, 1320]];next raw list=[66, 264, 660, 1320];all intermediate products SAFE

Higher derivative level3 ORIGINAL power1 coefficient product := mul(264, 1)

Higher derivative level3 ORIGINAL power1 coefficient product = 264

Higher derivative level3 ORIGINAL power2 coefficient product := mul(660, 2)

Higher derivative level3 ORIGINAL power2 coefficient product = 1320

Higher derivative level3 ORIGINAL power3 coefficient product := mul(1320, 3)

Higher derivative level3 ORIGINAL power3 coefficient product = 3960

previous derivative order=2;actual low-to-high list=[66, 264, 660, 1320];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 264, 1, 264], [2, 660, 2, 1320], [3, 1320, 3, 3960]];next raw list=[264, 1320, 3960];all intermediate products SAFE

ORIGINAL coefficients=[11, 22, 33, 44, 55, 66];order=3;declared degree bound=5

final derivative coefficients=[264, 1320, 3960];for each original power k>=order multiplier=k*(k-1)*...*(k-order+1);lower powers vanish

Higher derivative Horner power2 previous value times ORIGINAL point := mul(0, -2)

Higher derivative Horner power2 previous value times ORIGINAL point = 0

Higher derivative Horner power2 plus ORIGINAL derivative coefficient := sum(0, 3960)

Higher derivative Horner power2 plus ORIGINAL derivative coefficient = 3960

Higher derivative Horner power1 previous value times ORIGINAL point := mul(3960, -2)

Higher derivative Horner power1 previous value times ORIGINAL point = -7920

Higher derivative Horner power1 plus ORIGINAL derivative coefficient := sum(-7920, 1320)

Higher derivative Horner power1 plus ORIGINAL derivative coefficient = -6600

Higher derivative Horner power0 previous value times ORIGINAL point := mul(-6600, -2)

Higher derivative Horner power0 previous value times ORIGINAL point = 13200

Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(13200, 264)

Higher derivative Horner power0 plus ORIGINAL derivative coefficient = 13464

逐步反向核對

• S024 · 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。
• S023 · 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。
• S022 · 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。
• S021 · 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。
• S020 · 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。
• S019 · 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。
• 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 — 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

([11, 22, 33, 44, 55, 66], 3, [264, 1320, 3960], -2, 13464, 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。