第 23095 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":[20,40,60,80,100,120,140],"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":[20,40,60,80,100,120,140],"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=[20, 40, 60, 80, 100, 120, 140];declared order=8

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

Higher derivative level1 ORIGINAL power1 coefficient product := mul(40, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 40

Higher derivative level1 ORIGINAL power2 coefficient product := mul(60, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 120

Higher derivative level1 ORIGINAL power3 coefficient product := mul(80, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 240

Higher derivative level1 ORIGINAL power4 coefficient product := mul(100, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 400

Higher derivative level1 ORIGINAL power5 coefficient product := mul(120, 5)

Higher derivative level1 ORIGINAL power5 coefficient product = 600

Higher derivative level1 ORIGINAL power6 coefficient product := mul(140, 6)

Higher derivative level1 ORIGINAL power6 coefficient product = 840

previous derivative order=0;actual low-to-high list=[20, 40, 60, 80, 100, 120, 140];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 40, 1, 40], [2, 60, 2, 120], [3, 80, 3, 240], [4, 100, 4, 400], [5, 120, 5, 600], [6, 140, 6, 840]];next raw list=[40, 120, 240, 400, 600, 840];all intermediate products SAFE

Higher derivative level2 ORIGINAL power1 coefficient product := mul(120, 1)

Higher derivative level2 ORIGINAL power1 coefficient product = 120

Higher derivative level2 ORIGINAL power2 coefficient product := mul(240, 2)

Higher derivative level2 ORIGINAL power2 coefficient product = 480

Higher derivative level2 ORIGINAL power3 coefficient product := mul(400, 3)

Higher derivative level2 ORIGINAL power3 coefficient product = 1200

Higher derivative level2 ORIGINAL power4 coefficient product := mul(600, 4)

Higher derivative level2 ORIGINAL power4 coefficient product = 2400

Higher derivative level2 ORIGINAL power5 coefficient product := mul(840, 5)

Higher derivative level2 ORIGINAL power5 coefficient product = 4200

previous derivative order=1;actual low-to-high list=[40, 120, 240, 400, 600, 840];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 120, 1, 120], [2, 240, 2, 480], [3, 400, 3, 1200], [4, 600, 4, 2400], [5, 840, 5, 4200]];next raw list=[120, 480, 1200, 2400, 4200];all intermediate products SAFE

Higher derivative level3 ORIGINAL power1 coefficient product := mul(480, 1)

Higher derivative level3 ORIGINAL power1 coefficient product = 480

Higher derivative level3 ORIGINAL power2 coefficient product := mul(1200, 2)

Higher derivative level3 ORIGINAL power2 coefficient product = 2400

Higher derivative level3 ORIGINAL power3 coefficient product := mul(2400, 3)

Higher derivative level3 ORIGINAL power3 coefficient product = 7200

Higher derivative level3 ORIGINAL power4 coefficient product := mul(4200, 4)

Higher derivative level3 ORIGINAL power4 coefficient product = 16800

previous derivative order=2;actual low-to-high list=[120, 480, 1200, 2400, 4200];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 480, 1, 480], [2, 1200, 2, 2400], [3, 2400, 3, 7200], [4, 4200, 4, 16800]];next raw list=[480, 2400, 7200, 16800];all intermediate products SAFE

Higher derivative level4 ORIGINAL power1 coefficient product := mul(2400, 1)

Higher derivative level4 ORIGINAL power1 coefficient product = 2400

Higher derivative level4 ORIGINAL power2 coefficient product := mul(7200, 2)

Higher derivative level4 ORIGINAL power2 coefficient product = 14400

Higher derivative level4 ORIGINAL power3 coefficient product := mul(16800, 3)

Higher derivative level4 ORIGINAL power3 coefficient product = 50400

previous derivative order=3;actual low-to-high list=[480, 2400, 7200, 16800];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 2400, 1, 2400], [2, 7200, 2, 14400], [3, 16800, 3, 50400]];next raw list=[2400, 14400, 50400];all intermediate products SAFE

Higher derivative level5 ORIGINAL power1 coefficient product := mul(14400, 1)

Higher derivative level5 ORIGINAL power1 coefficient product = 14400

Higher derivative level5 ORIGINAL power2 coefficient product := mul(50400, 2)

Higher derivative level5 ORIGINAL power2 coefficient product = 100800

previous derivative order=4;actual low-to-high list=[2400, 14400, 50400];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 14400, 1, 14400], [2, 50400, 2, 100800]];next raw list=[14400, 100800];all intermediate products SAFE

Higher derivative level6 ORIGINAL power1 coefficient product := mul(100800, 1)

Higher derivative level6 ORIGINAL power1 coefficient product = 100800

previous derivative order=5;actual low-to-high list=[14400, 100800];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 100800, 1, 100800]];next raw list=[100800];all intermediate products SAFE

previous derivative order=6;actual low-to-high list=[100800];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=[20, 40, 60, 80, 100, 120, 140];order=8;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, 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

• S034 · 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。
• S033 · 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。
• 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, 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。