第 23087 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-zero-beyond-degree；資料={"coefficients":[12,24,36,48,60,72,84],"order":9,"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-zero-beyond-degree；資料={"coefficients":[12,24,36,48,60,72,84],"order":9,"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-zero-beyond-degree;original snapshot and mathematics retained

explicit ordered source output fields=[declared_degree_bound, order, derivative_coefficients, point, value, zero_derivative_certificate, certificate]

ORIGINAL low-to-high coefficients=[12, 24, 36, 48, 60, 72, 84];declared order=9

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

Higher derivative level1 ORIGINAL power1 coefficient product := mul(24, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 24

Higher derivative level1 ORIGINAL power2 coefficient product := mul(36, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 72

Higher derivative level1 ORIGINAL power3 coefficient product := mul(48, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 144

Higher derivative level1 ORIGINAL power4 coefficient product := mul(60, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 240

Higher derivative level1 ORIGINAL power5 coefficient product := mul(72, 5)

Higher derivative level1 ORIGINAL power5 coefficient product = 360

Higher derivative level1 ORIGINAL power6 coefficient product := mul(84, 6)

Higher derivative level1 ORIGINAL power6 coefficient product = 504

previous derivative order=0;actual low-to-high list=[12, 24, 36, 48, 60, 72, 84];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 24, 1, 24], [2, 36, 2, 72], [3, 48, 3, 144], [4, 60, 4, 240], [5, 72, 5, 360], [6, 84, 6, 504]];next raw list=[24, 72, 144, 240, 360, 504];all intermediate products SAFE

Higher derivative level2 ORIGINAL power1 coefficient product := mul(72, 1)

Higher derivative level2 ORIGINAL power1 coefficient product = 72

Higher derivative level2 ORIGINAL power2 coefficient product := mul(144, 2)

Higher derivative level2 ORIGINAL power2 coefficient product = 288

Higher derivative level2 ORIGINAL power3 coefficient product := mul(240, 3)

Higher derivative level2 ORIGINAL power3 coefficient product = 720

Higher derivative level2 ORIGINAL power4 coefficient product := mul(360, 4)

Higher derivative level2 ORIGINAL power4 coefficient product = 1440

Higher derivative level2 ORIGINAL power5 coefficient product := mul(504, 5)

Higher derivative level2 ORIGINAL power5 coefficient product = 2520

previous derivative order=1;actual low-to-high list=[24, 72, 144, 240, 360, 504];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 72, 1, 72], [2, 144, 2, 288], [3, 240, 3, 720], [4, 360, 4, 1440], [5, 504, 5, 2520]];next raw list=[72, 288, 720, 1440, 2520];all intermediate products SAFE

Higher derivative level3 ORIGINAL power1 coefficient product := mul(288, 1)

Higher derivative level3 ORIGINAL power1 coefficient product = 288

Higher derivative level3 ORIGINAL power2 coefficient product := mul(720, 2)

Higher derivative level3 ORIGINAL power2 coefficient product = 1440

Higher derivative level3 ORIGINAL power3 coefficient product := mul(1440, 3)

Higher derivative level3 ORIGINAL power3 coefficient product = 4320

Higher derivative level3 ORIGINAL power4 coefficient product := mul(2520, 4)

Higher derivative level3 ORIGINAL power4 coefficient product = 10080

previous derivative order=2;actual low-to-high list=[72, 288, 720, 1440, 2520];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 288, 1, 288], [2, 720, 2, 1440], [3, 1440, 3, 4320], [4, 2520, 4, 10080]];next raw list=[288, 1440, 4320, 10080];all intermediate products SAFE

Higher derivative level4 ORIGINAL power1 coefficient product := mul(1440, 1)

Higher derivative level4 ORIGINAL power1 coefficient product = 1440

Higher derivative level4 ORIGINAL power2 coefficient product := mul(4320, 2)

Higher derivative level4 ORIGINAL power2 coefficient product = 8640

Higher derivative level4 ORIGINAL power3 coefficient product := mul(10080, 3)

Higher derivative level4 ORIGINAL power3 coefficient product = 30240

previous derivative order=3;actual low-to-high list=[288, 1440, 4320, 10080];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 1440, 1, 1440], [2, 4320, 2, 8640], [3, 10080, 3, 30240]];next raw list=[1440, 8640, 30240];all intermediate products SAFE

Higher derivative level5 ORIGINAL power1 coefficient product := mul(8640, 1)

Higher derivative level5 ORIGINAL power1 coefficient product = 8640

Higher derivative level5 ORIGINAL power2 coefficient product := mul(30240, 2)

Higher derivative level5 ORIGINAL power2 coefficient product = 60480

previous derivative order=4;actual low-to-high list=[1440, 8640, 30240];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 8640, 1, 8640], [2, 30240, 2, 60480]];next raw list=[8640, 60480];all intermediate products SAFE

Higher derivative level6 ORIGINAL power1 coefficient product := mul(60480, 1)

Higher derivative level6 ORIGINAL power1 coefficient product = 60480

previous derivative order=5;actual low-to-high list=[8640, 60480];drop ONLY its constant term

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

previous derivative order=6;actual low-to-high list=[60480];drop ONLY its constant term

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

previous derivative order=7;actual low-to-high list=[];drop ONLY its constant term

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

previous derivative order=8;actual low-to-high list=[];drop ONLY its constant term

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

ORIGINAL coefficients=[12, 24, 36, 48, 60, 72, 84];order=9;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

• S035 · 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。
• S034 · 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。
• S033 · 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。
• S032 · 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。
• 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

(6, 9, [0], -1, 0, 是, orders_above_polynomial_degree_are_zero_without_analyticity_claim)

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。