第 23069 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":[19,38,57,76,95,114],"order":4,"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":[19,38,57,76,95,114],"order":4,"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=[19, 38, 57, 76, 95, 114];declared order=4

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

Higher derivative level1 ORIGINAL power1 coefficient product := mul(38, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 38

Higher derivative level1 ORIGINAL power2 coefficient product := mul(57, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 114

Higher derivative level1 ORIGINAL power3 coefficient product := mul(76, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 228

Higher derivative level1 ORIGINAL power4 coefficient product := mul(95, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 380

Higher derivative level1 ORIGINAL power5 coefficient product := mul(114, 5)

Higher derivative level1 ORIGINAL power5 coefficient product = 570

previous derivative order=0;actual low-to-high list=[19, 38, 57, 76, 95, 114];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 38, 1, 38], [2, 57, 2, 114], [3, 76, 3, 228], [4, 95, 4, 380], [5, 114, 5, 570]];next raw list=[38, 114, 228, 380, 570];all intermediate products SAFE

Higher derivative level2 ORIGINAL power1 coefficient product := mul(114, 1)

Higher derivative level2 ORIGINAL power1 coefficient product = 114

Higher derivative level2 ORIGINAL power2 coefficient product := mul(228, 2)

Higher derivative level2 ORIGINAL power2 coefficient product = 456

Higher derivative level2 ORIGINAL power3 coefficient product := mul(380, 3)

Higher derivative level2 ORIGINAL power3 coefficient product = 1140

Higher derivative level2 ORIGINAL power4 coefficient product := mul(570, 4)

Higher derivative level2 ORIGINAL power4 coefficient product = 2280

previous derivative order=1;actual low-to-high list=[38, 114, 228, 380, 570];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 114, 1, 114], [2, 228, 2, 456], [3, 380, 3, 1140], [4, 570, 4, 2280]];next raw list=[114, 456, 1140, 2280];all intermediate products SAFE

Higher derivative level3 ORIGINAL power1 coefficient product := mul(456, 1)

Higher derivative level3 ORIGINAL power1 coefficient product = 456

Higher derivative level3 ORIGINAL power2 coefficient product := mul(1140, 2)

Higher derivative level3 ORIGINAL power2 coefficient product = 2280

Higher derivative level3 ORIGINAL power3 coefficient product := mul(2280, 3)

Higher derivative level3 ORIGINAL power3 coefficient product = 6840

previous derivative order=2;actual low-to-high list=[114, 456, 1140, 2280];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 456, 1, 456], [2, 1140, 2, 2280], [3, 2280, 3, 6840]];next raw list=[456, 2280, 6840];all intermediate products SAFE

Higher derivative level4 ORIGINAL power1 coefficient product := mul(2280, 1)

Higher derivative level4 ORIGINAL power1 coefficient product = 2280

Higher derivative level4 ORIGINAL power2 coefficient product := mul(6840, 2)

Higher derivative level4 ORIGINAL power2 coefficient product = 13680

previous derivative order=3;actual low-to-high list=[456, 2280, 6840];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 2280, 1, 2280], [2, 6840, 2, 13680]];next raw list=[2280, 13680];all intermediate products SAFE

ORIGINAL coefficients=[19, 38, 57, 76, 95, 114];order=4;declared degree bound=5

final derivative coefficients=[2280, 13680];for each original power k>=order multiplier=k*(k-1)*...*(k-order+1);lower powers vanish

Higher derivative Horner power1 previous value times ORIGINAL point := mul(0, 1)

Higher derivative Horner power1 previous value times ORIGINAL point = 0

Higher derivative Horner power1 plus ORIGINAL derivative coefficient := sum(0, 13680)

Higher derivative Horner power1 plus ORIGINAL derivative coefficient = 13680

Higher derivative Horner power0 previous value times ORIGINAL point := mul(13680, 1)

Higher derivative Horner power0 previous value times ORIGINAL point = 13680

Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(13680, 2280)

Higher derivative Horner power0 plus ORIGINAL derivative coefficient = 15960

逐步反向核對

• S025 · 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。
• S024 · 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。
• S023 · 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。
• S022 · 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。
• S021 · 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。
• 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

([19, 38, 57, 76, 95, 114], 4, [2280, 13680], 1, 15960, 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。