第 23090 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":[15,30,45,60,75,90],"order":8,"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-zero-beyond-degree；資料={"coefficients":[15,30,45,60,75,90],"order":8,"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-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=[15, 30, 45, 60, 75, 90];declared order=8

declared degree BOUND=5=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

Higher derivative level1 ORIGINAL power1 coefficient product := mul(30, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 30

Higher derivative level1 ORIGINAL power2 coefficient product := mul(45, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 90

Higher derivative level1 ORIGINAL power3 coefficient product := mul(60, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 180

Higher derivative level1 ORIGINAL power4 coefficient product := mul(75, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 300

Higher derivative level1 ORIGINAL power5 coefficient product := mul(90, 5)

Higher derivative level1 ORIGINAL power5 coefficient product = 450

previous derivative order=0;actual low-to-high list=[15, 30, 45, 60, 75, 90];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 30, 1, 30], [2, 45, 2, 90], [3, 60, 3, 180], [4, 75, 4, 300], [5, 90, 5, 450]];next raw list=[30, 90, 180, 300, 450];all intermediate products SAFE

Higher derivative level2 ORIGINAL power1 coefficient product := mul(90, 1)

Higher derivative level2 ORIGINAL power1 coefficient product = 90

Higher derivative level2 ORIGINAL power2 coefficient product := mul(180, 2)

Higher derivative level2 ORIGINAL power2 coefficient product = 360

Higher derivative level2 ORIGINAL power3 coefficient product := mul(300, 3)

Higher derivative level2 ORIGINAL power3 coefficient product = 900

Higher derivative level2 ORIGINAL power4 coefficient product := mul(450, 4)

Higher derivative level2 ORIGINAL power4 coefficient product = 1800

previous derivative order=1;actual low-to-high list=[30, 90, 180, 300, 450];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 90, 1, 90], [2, 180, 2, 360], [3, 300, 3, 900], [4, 450, 4, 1800]];next raw list=[90, 360, 900, 1800];all intermediate products SAFE

Higher derivative level3 ORIGINAL power1 coefficient product := mul(360, 1)

Higher derivative level3 ORIGINAL power1 coefficient product = 360

Higher derivative level3 ORIGINAL power2 coefficient product := mul(900, 2)

Higher derivative level3 ORIGINAL power2 coefficient product = 1800

Higher derivative level3 ORIGINAL power3 coefficient product := mul(1800, 3)

Higher derivative level3 ORIGINAL power3 coefficient product = 5400

previous derivative order=2;actual low-to-high list=[90, 360, 900, 1800];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 360, 1, 360], [2, 900, 2, 1800], [3, 1800, 3, 5400]];next raw list=[360, 1800, 5400];all intermediate products SAFE

Higher derivative level4 ORIGINAL power1 coefficient product := mul(1800, 1)

Higher derivative level4 ORIGINAL power1 coefficient product = 1800

Higher derivative level4 ORIGINAL power2 coefficient product := mul(5400, 2)

Higher derivative level4 ORIGINAL power2 coefficient product = 10800

previous derivative order=3;actual low-to-high list=[360, 1800, 5400];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 1800, 1, 1800], [2, 5400, 2, 10800]];next raw list=[1800, 10800];all intermediate products SAFE

Higher derivative level5 ORIGINAL power1 coefficient product := mul(10800, 1)

Higher derivative level5 ORIGINAL power1 coefficient product = 10800

previous derivative order=4;actual low-to-high list=[1800, 10800];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 10800, 1, 10800]];next raw list=[10800];all intermediate products SAFE

previous derivative order=5;actual low-to-high list=[10800];drop ONLY its constant term

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

previous derivative order=6;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=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

ORIGINAL coefficients=[15, 30, 45, 60, 75, 90];order=8;declared degree bound=5

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, 2)

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

• S028 · 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。
• S027 · 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。
• S026 · 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。
• S025 · 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。
• 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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S022 · 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。
• S021 · 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。
• 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 level4 ORIGINAL power2 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 level4 ORIGINAL power1 coefficient product，不以小數近似替代。
• S017 · 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。
• S016 · 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。
• S015 · 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。
• S014 · 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。
• S013 · 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。
• S012 · 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。
• S011 · 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。
• S010 · 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。
• S009 · 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。
• S008 · 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。
• 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

(5, 8, [0], 2, 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。