{"schema_version":"ocme-site-results-v1","locale":"en","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"},"kind":"guides","total":2,"pages":1,"page":1,"entries":[{"id":"derivation-mko-derivative-01","title":{"zh-Hant":"導數：由差商極限求二次式導數並判定尖點無導數","en":"The derivative: a quadratic's derivative from the difference-quotient limit, and no derivative at a corner"},"summary":{"zh-Hant":"對 f(t)=3t²−2t+5：先對 h≠0 化簡差商 (f(1+h)−f(1))/h=4+3h，再以 ε-δ 證明 f'(1)=4；推導一般公式：差商為 2a2x+a1+a2h、導數為 2a2x+a1；說明有限步 h=1/10 的差商 4.3 不是導數、a2=0 的仿射邊界；最後證明 g(t)=|t−2| 在 t=2 沒有導數（不是 0），而在 t>2 處導數為 1。","en":"For f(t)=3t²−2t+5: first simplify the difference quotient for h≠0, (f(1+h)−f(1))/h=4+3h, then prove f'(1)=4 by ε-δ; derive the general formulas: the quotient is 2a2x+a1+a2h and the derivative 2a2x+a1; explain why the finite-step quotient 4.3 at h=1/10 is not the derivative, and the affine boundary a2=0; finally prove that g(t)=|t−2| has no derivative at t=2 (not 0), while its derivative is 1 for t>2."},"mko_ids":["mko-derivative"]},{"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":"Concept derivation guides"}