第 23074 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":[24,48,72,96,120,144,168],"order":7,"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":[24,48,72,96,120,144,168],"order":7,"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=[24, 48, 72, 96, 120, 144, 168];declared order=7

declared degree BOUND=6=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

Higher derivative level1 ORIGINAL power1 coefficient product := mul(48, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 48

Higher derivative level1 ORIGINAL power2 coefficient product := mul(72, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 144

Higher derivative level1 ORIGINAL power3 coefficient product := mul(96, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 288

Higher derivative level1 ORIGINAL power4 coefficient product := mul(120, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 480

Higher derivative level1 ORIGINAL power5 coefficient product := mul(144, 5)

Higher derivative level1 ORIGINAL power5 coefficient product = 720

Higher derivative level1 ORIGINAL power6 coefficient product := mul(168, 6)

Higher derivative level1 ORIGINAL power6 coefficient product = 1008

previous derivative order=0;actual low-to-high list=[24, 48, 72, 96, 120, 144, 168];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 48, 1, 48], [2, 72, 2, 144], [3, 96, 3, 288], [4, 120, 4, 480], [5, 144, 5, 720], [6, 168, 6, 1008]];next raw list=[48, 144, 288, 480, 720, 1008];all intermediate products SAFE

Higher derivative level2 ORIGINAL power1 coefficient product := mul(144, 1)

Higher derivative level2 ORIGINAL power1 coefficient product = 144

Higher derivative level2 ORIGINAL power2 coefficient product := mul(288, 2)

Higher derivative level2 ORIGINAL power2 coefficient product = 576

Higher derivative level2 ORIGINAL power3 coefficient product := mul(480, 3)

Higher derivative level2 ORIGINAL power3 coefficient product = 1440

Higher derivative level2 ORIGINAL power4 coefficient product := mul(720, 4)

Higher derivative level2 ORIGINAL power4 coefficient product = 2880

Higher derivative level2 ORIGINAL power5 coefficient product := mul(1008, 5)

Higher derivative level2 ORIGINAL power5 coefficient product = 5040

previous derivative order=1;actual low-to-high list=[48, 144, 288, 480, 720, 1008];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 144, 1, 144], [2, 288, 2, 576], [3, 480, 3, 1440], [4, 720, 4, 2880], [5, 1008, 5, 5040]];next raw list=[144, 576, 1440, 2880, 5040];all intermediate products SAFE

Higher derivative level3 ORIGINAL power1 coefficient product := mul(576, 1)

Higher derivative level3 ORIGINAL power1 coefficient product = 576

Higher derivative level3 ORIGINAL power2 coefficient product := mul(1440, 2)

Higher derivative level3 ORIGINAL power2 coefficient product = 2880

Higher derivative level3 ORIGINAL power3 coefficient product := mul(2880, 3)

Higher derivative level3 ORIGINAL power3 coefficient product = 8640

Higher derivative level3 ORIGINAL power4 coefficient product := mul(5040, 4)

Higher derivative level3 ORIGINAL power4 coefficient product = 20160

previous derivative order=2;actual low-to-high list=[144, 576, 1440, 2880, 5040];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 576, 1, 576], [2, 1440, 2, 2880], [3, 2880, 3, 8640], [4, 5040, 4, 20160]];next raw list=[576, 2880, 8640, 20160];all intermediate products SAFE

Higher derivative level4 ORIGINAL power1 coefficient product := mul(2880, 1)

Higher derivative level4 ORIGINAL power1 coefficient product = 2880

Higher derivative level4 ORIGINAL power2 coefficient product := mul(8640, 2)

Higher derivative level4 ORIGINAL power2 coefficient product = 17280

Higher derivative level4 ORIGINAL power3 coefficient product := mul(20160, 3)

Higher derivative level4 ORIGINAL power3 coefficient product = 60480

previous derivative order=3;actual low-to-high list=[576, 2880, 8640, 20160];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 2880, 1, 2880], [2, 8640, 2, 17280], [3, 20160, 3, 60480]];next raw list=[2880, 17280, 60480];all intermediate products SAFE

Higher derivative level5 ORIGINAL power1 coefficient product := mul(17280, 1)

Higher derivative level5 ORIGINAL power1 coefficient product = 17280

Higher derivative level5 ORIGINAL power2 coefficient product := mul(60480, 2)

Higher derivative level5 ORIGINAL power2 coefficient product = 120960

previous derivative order=4;actual low-to-high list=[2880, 17280, 60480];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 17280, 1, 17280], [2, 60480, 2, 120960]];next raw list=[17280, 120960];all intermediate products SAFE

Higher derivative level6 ORIGINAL power1 coefficient product := mul(120960, 1)

Higher derivative level6 ORIGINAL power1 coefficient product = 120960

previous derivative order=5;actual low-to-high list=[17280, 120960];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 120960, 1, 120960]];next raw list=[120960];all intermediate products SAFE

previous derivative order=6;actual low-to-high list=[120960];drop ONLY its constant term

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

ORIGINAL coefficients=[24, 48, 72, 96, 120, 144, 168];order=7;declared degree bound=6

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

• S033 · 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。
• S032 · 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。
• S031 · 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。
• S030 · 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。
• S029 · 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。
• S028 · premise_and_result_audit — 以精確有理數重算 Higher derivative level6 ORIGINAL power1 coefficient product，不以小數近似替代。
• S027 · 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。
• S026 · premise_and_result_audit — 以精確有理數重算 Higher derivative level5 ORIGINAL power2 coefficient product，不以小數近似替代。
• S025 · 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。
• 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 — 以精確有理數重算 Higher derivative level4 ORIGINAL power3 coefficient product，不以小數近似替代。
• S022 · 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。
• S021 · 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。
• 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 level3 ORIGINAL power4 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 level3 ORIGINAL power3 coefficient product，不以小數近似替代。
• S017 · 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。
• S016 · 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。
• S015 · 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。
• S014 · premise_and_result_audit — 以精確有理數重算 Higher derivative level2 ORIGINAL power5 coefficient product，不以小數近似替代。
• S013 · 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。
• S012 · 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。
• S011 · 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。
• S010 · 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。
• 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 level1 ORIGINAL power6 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 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

([24, 48, 72, 96, 120, 144, 168], 7, [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。