第 31171 題：多項式乘積的導數

原題與已知條件

全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[8, 21, 1, 1]、g=[2, -1, 1, 4]，兩輸入多項式各至多三次；求 fg 在 x=-2 的導數值。

• G001 — 全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[8, 21, 1, 1]、g=[2, -1, 1, 4]，兩輸入多項式各至多三次；求 fg 在 x=-2 的導數值。

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-derivative-rules

逐步推導

source roles = [qf-derivative-rule-product, derivative_product, domain, reals]

derived = [[integer_derivative_value]]

source roles = [f OUTER or first, [8, 21, 1, 1]]

derived = [[8, 21, 1, 1]]

source roles = [g INNER or second, [2, -1, 1, 4]]

derived = [[2, -1, 1, 4]]

f derivative coefficient power1 := mul(1, 21)

f derivative coefficient power1 = 21

f derivative coefficient power2 := mul(2, 1)

f derivative coefficient power2 = 2

f derivative coefficient power3 := mul(3, 1)

f derivative coefficient power3 = 3

source roles = [f, [8, 21, 1, 1]]

derived = [[21, 2, 3]]

g derivative coefficient power1 := mul(1, -1)

g derivative coefficient power1 = -1

g derivative coefficient power2 := mul(2, 1)

g derivative coefficient power2 = 2

g derivative coefficient power3 := mul(3, 4)

g derivative coefficient power3 = 12

source roles = [g, [2, -1, 1, 4]]

derived = [[-1, 2, 12]]

g derivative at ORIGINAL x Horner power2 times argument := mul(0, -2)

g derivative at ORIGINAL x Horner power2 times argument = 0

g derivative at ORIGINAL x Horner power2 plus coefficient := sum(0, 12)

g derivative at ORIGINAL x Horner power2 plus coefficient = 12

g derivative at ORIGINAL x Horner power1 times argument := mul(12, -2)

g derivative at ORIGINAL x Horner power1 times argument = -24

g derivative at ORIGINAL x Horner power1 plus coefficient := sum(-24, 2)

g derivative at ORIGINAL x Horner power1 plus coefficient = -22

g derivative at ORIGINAL x Horner power0 times argument := mul(-22, -2)

g derivative at ORIGINAL x Horner power0 times argument = 44

g derivative at ORIGINAL x Horner power0 plus coefficient := sum(44, -1)

g derivative at ORIGINAL x Horner power0 plus coefficient = 43

source roles = [g derivative at ORIGINAL x, [-1, 2, 12], -2]

derived = [43]

f derivative at ORIGINAL x Horner power2 times argument := mul(0, -2)

f derivative at ORIGINAL x Horner power2 times argument = 0

f derivative at ORIGINAL x Horner power2 plus coefficient := sum(0, 3)

f derivative at ORIGINAL x Horner power2 plus coefficient = 3

f derivative at ORIGINAL x Horner power1 times argument := mul(3, -2)

f derivative at ORIGINAL x Horner power1 times argument = -6

f derivative at ORIGINAL x Horner power1 plus coefficient := sum(-6, 2)

f derivative at ORIGINAL x Horner power1 plus coefficient = -4

f derivative at ORIGINAL x Horner power0 times argument := mul(-4, -2)

f derivative at ORIGINAL x Horner power0 times argument = 8

f derivative at ORIGINAL x Horner power0 plus coefficient := sum(8, 21)

f derivative at ORIGINAL x Horner power0 plus coefficient = 29

source roles = [f derivative at ORIGINAL x, [21, 2, 3], -2]

derived = [29]

f value at ORIGINAL x Horner power3 times argument := mul(0, -2)

f value at ORIGINAL x Horner power3 times argument = 0

f value at ORIGINAL x Horner power3 plus coefficient := sum(0, 1)

f value at ORIGINAL x Horner power3 plus coefficient = 1

f value at ORIGINAL x Horner power2 times argument := mul(1, -2)

f value at ORIGINAL x Horner power2 times argument = -2

f value at ORIGINAL x Horner power2 plus coefficient := sum(-2, 1)

f value at ORIGINAL x Horner power2 plus coefficient = -1

f value at ORIGINAL x Horner power1 times argument := mul(-1, -2)

f value at ORIGINAL x Horner power1 times argument = 2

f value at ORIGINAL x Horner power1 plus coefficient := sum(2, 21)

f value at ORIGINAL x Horner power1 plus coefficient = 23

f value at ORIGINAL x Horner power0 times argument := mul(23, -2)

f value at ORIGINAL x Horner power0 times argument = -46

f value at ORIGINAL x Horner power0 plus coefficient := sum(-46, 8)

f value at ORIGINAL x Horner power0 plus coefficient = -38

source roles = [f value at ORIGINAL x, [8, 21, 1, 1], -2]

derived = [-38]

g value at ORIGINAL x Horner power3 times argument := mul(0, -2)

g value at ORIGINAL x Horner power3 times argument = 0

g value at ORIGINAL x Horner power3 plus coefficient := sum(0, 4)

g value at ORIGINAL x Horner power3 plus coefficient = 4

g value at ORIGINAL x Horner power2 times argument := mul(4, -2)

g value at ORIGINAL x Horner power2 times argument = -8

g value at ORIGINAL x Horner power2 plus coefficient := sum(-8, 1)

g value at ORIGINAL x Horner power2 plus coefficient = -7

g value at ORIGINAL x Horner power1 times argument := mul(-7, -2)

g value at ORIGINAL x Horner power1 times argument = 14

g value at ORIGINAL x Horner power1 plus coefficient := sum(14, -1)

g value at ORIGINAL x Horner power1 plus coefficient = 13

g value at ORIGINAL x Horner power0 times argument := mul(13, -2)

g value at ORIGINAL x Horner power0 times argument = -26

g value at ORIGINAL x Horner power0 plus coefficient := sum(-26, 2)

g value at ORIGINAL x Horner power0 plus coefficient = -24

source roles = [g value at ORIGINAL x, [2, -1, 1, 4], -2]

derived = [-24]

Derivative product ORIGINAL fprime times g := mul(29, -24)

Derivative product ORIGINAL fprime times g = -696

Derivative product ORIGINAL f times gprime := mul(-38, 43)

Derivative product ORIGINAL f times gprime = -1634

Derivative product ORIGINAL two terms sum := sum(-696, -1634)

Derivative product ORIGINAL two terms sum = -2330

source roles = [[8, 21, 1, 1], [2, -1, 1, 4], -2, [21, 2, 3], [-1, 2, 12]]

derived = [-38, -24, -696, -1634, -2330]

逐步反向核對

• S047 · premise_and_result_audit — 以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。
• S046 · premise_and_result_audit — 以精確有理數重算 Derivative product ORIGINAL two terms sum，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Derivative product ORIGINAL f times gprime，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Derivative product ORIGINAL fprime times g，不以小數近似替代。
• S043 · premise_and_result_audit — 以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。
• S042 · premise_and_result_audit — 以精確有理數重算 g value at ORIGINAL x Horner power0 plus coefficient，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 g value at ORIGINAL x Horner power0 times argument，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 g value at ORIGINAL x Horner power1 plus coefficient，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 g value at ORIGINAL x Horner power1 times argument，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 g value at ORIGINAL x Horner power2 plus coefficient，不以小數近似替代。
• S037 · premise_and_result_audit — 以精確有理數重算 g value at ORIGINAL x Horner power2 times argument，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 g value at ORIGINAL x Horner power3 plus coefficient，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 g value at ORIGINAL x Horner power3 times argument，不以小數近似替代。
• S034 · premise_and_result_audit — 以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。
• S033 · premise_and_result_audit — 以精確有理數重算 f value at ORIGINAL x Horner power0 plus coefficient，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 f value at ORIGINAL x Horner power0 times argument，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 f value at ORIGINAL x Horner power1 plus coefficient，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 f value at ORIGINAL x Horner power1 times argument，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 f value at ORIGINAL x Horner power2 plus coefficient，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 f value at ORIGINAL x Horner power2 times argument，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 f value at ORIGINAL x Horner power3 plus coefficient，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 f value at ORIGINAL x Horner power3 times argument，不以小數近似替代。
• S025 · premise_and_result_audit — 以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。
• S024 · premise_and_result_audit — 以精確有理數重算 f derivative at ORIGINAL x Horner power0 plus coefficient，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 f derivative at ORIGINAL x Horner power0 times argument，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 f derivative at ORIGINAL x Horner power1 plus coefficient，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 f derivative at ORIGINAL x Horner power1 times argument，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 f derivative at ORIGINAL x Horner power2 plus coefficient，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 f derivative at ORIGINAL x Horner power2 times argument，不以小數近似替代。
• S018 · premise_and_result_audit — 以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。
• S017 · premise_and_result_audit — 以精確有理數重算 g derivative at ORIGINAL x Horner power0 plus coefficient，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 g derivative at ORIGINAL x Horner power0 times argument，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 g derivative at ORIGINAL x Horner power1 plus coefficient，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 g derivative at ORIGINAL x Horner power1 times argument，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 g derivative at ORIGINAL x Horner power2 plus coefficient，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 g derivative at ORIGINAL x Horner power2 times argument，不以小數近似替代。
• S011 · premise_and_result_audit — 以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。
• S010 · premise_and_result_audit — 以精確有理數重算 g derivative coefficient power3，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 g derivative coefficient power2，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 g derivative coefficient power1，不以小數近似替代。
• S007 · premise_and_result_audit — 以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。
• S006 · premise_and_result_audit — 以精確有理數重算 f derivative coefficient power3，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 f derivative coefficient power2，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 f derivative coefficient power1，不以小數近似替代。
• S003 · premise_and_result_audit — 以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。
• S002 · premise_and_result_audit — 以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。
• S001 · premise_and_result_audit — 以原始輸入、型別及本步條件反向代回核對，不聲稱結論能唯一還原所有前提。這是來源綁定候選，不是獨立審查或形式證明。

結論

-2330

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。