{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-derivative-rules"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-derivative-rules-01","title":{"zh-Hant":"微分法則：以展開求導核對乘積與鏈式規則","en":"Derivative rules: checking the product and chain rules against differentiating the expansion"},"summary":{"zh-Hant":"以升冪係數列表 f=[2,0,1]、g=[1,3]（即 f(x)=2+x²、g(x)=1+3x）：先由導數定義證明三次以下多項式的求導公式；再以兩條獨立路徑求 (fg)' 與 (f∘g)'——展開後求導，以及使用乘積、鏈式規則——並確認兩者對每個 x 相同；同時核對和與純量倍的規則，指出 f'g'、漏乘內導數、外導數取錯點、反讀係數等錯誤，最後求 x=1 的值。","en":"With the ascending coefficient lists f=[2,0,1] and g=[1,3] (that is f(x)=2+x² and g(x)=1+3x): first prove from the definition of the derivative the differentiation formula for polynomials of degree at most three; then find (fg)' and (f∘g)' along two independent routes, differentiating the expansion and applying the product and chain rules, and confirm they agree at every x; also check the sum and scalar rules, expose the errors f'g', a missing inner derivative, an outer derivative at the wrong point and reversed coefficient reading, and evaluate at x=1."},"mko_ids":["mko-derivative-rules"]}],"title":"概念推導導讀"}