第 23054 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":[4,8,12,16,20,24,28],"order":3,"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":[4,8,12,16,20,24,28],"order":3,"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=[4, 8, 12, 16, 20, 24, 28];declared order=3

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

Higher derivative level1 ORIGINAL power1 coefficient product := mul(8, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 8

Higher derivative level1 ORIGINAL power2 coefficient product := mul(12, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 24

Higher derivative level1 ORIGINAL power3 coefficient product := mul(16, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 48

Higher derivative level1 ORIGINAL power4 coefficient product := mul(20, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 80

Higher derivative level1 ORIGINAL power5 coefficient product := mul(24, 5)

Higher derivative level1 ORIGINAL power5 coefficient product = 120

Higher derivative level1 ORIGINAL power6 coefficient product := mul(28, 6)

Higher derivative level1 ORIGINAL power6 coefficient product = 168

previous derivative order=0;actual low-to-high list=[4, 8, 12, 16, 20, 24, 28];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 8, 1, 8], [2, 12, 2, 24], [3, 16, 3, 48], [4, 20, 4, 80], [5, 24, 5, 120], [6, 28, 6, 168]];next raw list=[8, 24, 48, 80, 120, 168];all intermediate products SAFE

Higher derivative level2 ORIGINAL power1 coefficient product := mul(24, 1)

Higher derivative level2 ORIGINAL power1 coefficient product = 24

Higher derivative level2 ORIGINAL power2 coefficient product := mul(48, 2)

Higher derivative level2 ORIGINAL power2 coefficient product = 96

Higher derivative level2 ORIGINAL power3 coefficient product := mul(80, 3)

Higher derivative level2 ORIGINAL power3 coefficient product = 240

Higher derivative level2 ORIGINAL power4 coefficient product := mul(120, 4)

Higher derivative level2 ORIGINAL power4 coefficient product = 480

Higher derivative level2 ORIGINAL power5 coefficient product := mul(168, 5)

Higher derivative level2 ORIGINAL power5 coefficient product = 840

previous derivative order=1;actual low-to-high list=[8, 24, 48, 80, 120, 168];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 24, 1, 24], [2, 48, 2, 96], [3, 80, 3, 240], [4, 120, 4, 480], [5, 168, 5, 840]];next raw list=[24, 96, 240, 480, 840];all intermediate products SAFE

Higher derivative level3 ORIGINAL power1 coefficient product := mul(96, 1)

Higher derivative level3 ORIGINAL power1 coefficient product = 96

Higher derivative level3 ORIGINAL power2 coefficient product := mul(240, 2)

Higher derivative level3 ORIGINAL power2 coefficient product = 480

Higher derivative level3 ORIGINAL power3 coefficient product := mul(480, 3)

Higher derivative level3 ORIGINAL power3 coefficient product = 1440

Higher derivative level3 ORIGINAL power4 coefficient product := mul(840, 4)

Higher derivative level3 ORIGINAL power4 coefficient product = 3360

previous derivative order=2;actual low-to-high list=[24, 96, 240, 480, 840];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 96, 1, 96], [2, 240, 2, 480], [3, 480, 3, 1440], [4, 840, 4, 3360]];next raw list=[96, 480, 1440, 3360];all intermediate products SAFE

ORIGINAL coefficients=[4, 8, 12, 16, 20, 24, 28];order=3;declared degree bound=6

final derivative coefficients=[96, 480, 1440, 3360];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, 3360)

Higher derivative Horner power3 plus ORIGINAL derivative coefficient = 3360

Higher derivative Horner power2 previous value times ORIGINAL point := mul(3360, 1)

Higher derivative Horner power2 previous value times ORIGINAL point = 3360

Higher derivative Horner power2 plus ORIGINAL derivative coefficient := sum(3360, 1440)

Higher derivative Horner power2 plus ORIGINAL derivative coefficient = 4800

Higher derivative Horner power1 previous value times ORIGINAL point := mul(4800, 1)

Higher derivative Horner power1 previous value times ORIGINAL point = 4800

Higher derivative Horner power1 plus ORIGINAL derivative coefficient := sum(4800, 480)

Higher derivative Horner power1 plus ORIGINAL derivative coefficient = 5280

Higher derivative Horner power0 previous value times ORIGINAL point := mul(5280, 1)

Higher derivative Horner power0 previous value times ORIGINAL point = 5280

Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(5280, 96)

Higher derivative Horner power0 plus ORIGINAL derivative coefficient = 5376

逐步反向核對

• S029 · 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。
• S028 · 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。
• S027 · 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。
• S026 · 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。
• S025 · 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。
• S024 · 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。
• S023 · 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。
• S022 · 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。
• 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 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

([4, 8, 12, 16, 20, 24, 28], 3, [96, 480, 1440, 3360], 1, 5376, 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。