第 23071 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":[21,42,63,84],"order":0,"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":[21,42,63,84],"order":0,"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=[21, 42, 63, 84];declared order=0

declared degree BOUND=3=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

ORIGINAL coefficients=[21, 42, 63, 84];order=0;declared degree bound=3

final derivative coefficients=[21, 42, 63, 84];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, -2)

Higher derivative Horner power3 previous value times ORIGINAL point = 0

Higher derivative Horner power3 plus ORIGINAL derivative coefficient := sum(0, 84)

Higher derivative Horner power3 plus ORIGINAL derivative coefficient = 84

Higher derivative Horner power2 previous value times ORIGINAL point := mul(84, -2)

Higher derivative Horner power2 previous value times ORIGINAL point = -168

Higher derivative Horner power2 plus ORIGINAL derivative coefficient := sum(-168, 63)

Higher derivative Horner power2 plus ORIGINAL derivative coefficient = -105

Higher derivative Horner power1 previous value times ORIGINAL point := mul(-105, -2)

Higher derivative Horner power1 previous value times ORIGINAL point = 210

Higher derivative Horner power1 plus ORIGINAL derivative coefficient := sum(210, 42)

Higher derivative Horner power1 plus ORIGINAL derivative coefficient = 252

Higher derivative Horner power0 previous value times ORIGINAL point := mul(252, -2)

Higher derivative Horner power0 previous value times ORIGINAL point = -504

Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(-504, 21)

Higher derivative Horner power0 plus ORIGINAL derivative coefficient = -483

逐步反向核對

• S011 · 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。
• S010 · 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。
• S009 · 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。
• S008 · 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。
• S007 · 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。
• S006 · 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。
• S005 · 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。
• S004 · 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。
• S003 · 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。
• 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

([21, 42, 63, 84], 0, [21, 42, 63, 84], -2, -483, 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。