第 23060 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":[10,20,30,40,50],"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":[10,20,30,40,50],"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=[10, 20, 30, 40, 50];declared order=3

declared degree BOUND=4=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

Higher derivative level1 ORIGINAL power1 coefficient product := mul(20, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 20

Higher derivative level1 ORIGINAL power2 coefficient product := mul(30, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 60

Higher derivative level1 ORIGINAL power3 coefficient product := mul(40, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 120

Higher derivative level1 ORIGINAL power4 coefficient product := mul(50, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 200

previous derivative order=0;actual low-to-high list=[10, 20, 30, 40, 50];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 20, 1, 20], [2, 30, 2, 60], [3, 40, 3, 120], [4, 50, 4, 200]];next raw list=[20, 60, 120, 200];all intermediate products SAFE

Higher derivative level2 ORIGINAL power1 coefficient product := mul(60, 1)

Higher derivative level2 ORIGINAL power1 coefficient product = 60

Higher derivative level2 ORIGINAL power2 coefficient product := mul(120, 2)

Higher derivative level2 ORIGINAL power2 coefficient product = 240

Higher derivative level2 ORIGINAL power3 coefficient product := mul(200, 3)

Higher derivative level2 ORIGINAL power3 coefficient product = 600

previous derivative order=1;actual low-to-high list=[20, 60, 120, 200];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 60, 1, 60], [2, 120, 2, 240], [3, 200, 3, 600]];next raw list=[60, 240, 600];all intermediate products SAFE

Higher derivative level3 ORIGINAL power1 coefficient product := mul(240, 1)

Higher derivative level3 ORIGINAL power1 coefficient product = 240

Higher derivative level3 ORIGINAL power2 coefficient product := mul(600, 2)

Higher derivative level3 ORIGINAL power2 coefficient product = 1200

previous derivative order=2;actual low-to-high list=[60, 240, 600];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 240, 1, 240], [2, 600, 2, 1200]];next raw list=[240, 1200];all intermediate products SAFE

ORIGINAL coefficients=[10, 20, 30, 40, 50];order=3;declared degree bound=4

final derivative coefficients=[240, 1200];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, 2)

Higher derivative Horner power1 previous value times ORIGINAL point = 0

Higher derivative Horner power1 plus ORIGINAL derivative coefficient := sum(0, 1200)

Higher derivative Horner power1 plus ORIGINAL derivative coefficient = 1200

Higher derivative Horner power0 previous value times ORIGINAL point := mul(1200, 2)

Higher derivative Horner power0 previous value times ORIGINAL point = 2400

Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(2400, 240)

Higher derivative Horner power0 plus ORIGINAL derivative coefficient = 2640

逐步反向核對

• S019 · 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。
• S018 · 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。
• S017 · 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。
• S016 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S013 · 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。
• S012 · 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。
• S011 · 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。
• S010 · 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。
• S009 · 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。
• S008 · 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。
• S007 · 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。
• 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

([10, 20, 30, 40, 50], 3, [240, 1200], 2, 2640, 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。