{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-higher-order-derivative-01",
  "mko_id": "mko-higher-order-derivative",
  "title": {
    "zh-Hant": "四次多項式的全部導數階",
    "en": "Every derivative order of a quartic"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-higher-order-derivative-01",
    "source": {
      "mko_id": "mko-higher-order-derivative",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "78b36c7cb74b3af2a4e2f3cbb7c42f790660d13aa1347fdd50826cc718717741",
      "english_sha256": "c1886e7e00a610edfc2e3a3fb75552d4afb78b49f57c24e7e1abd7d5ce5775d6"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "polynomial",
      "algebra",
      "definition_here"
    ],
    "goal": {
      "zh-Hant": "由低到高係數 [2,-3,0,5,-2] 計算所有階導數。",
      "en": "Compute all derivative orders for low-to-high coefficients [2,-3,0,5,-2]."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "f(x)=2-3x+5x^3-2x^4，定義域 R，階數 k∈N_0。",
          "en": "f(x)=2-3x+5x^3-2x^4 on R, with derivative order k in N_0."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "f^{(0)}",
        "after_tex": "f^{(0)}(x)=f(x)=2-3x+5x^3-2x^4",
        "relation": "definition",
        "rule": "definition_here",
        "conditions": [],
        "explanation": {
          "zh-Hant": "零階導數按慣例是原函數，係數陣列中二次項確為零。",
          "en": "Order zero returns the original function; the coefficient array's quadratic term is zero."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "不能把零階誤作零函數。",
            "en": "Order zero must not be mistaken for the zero function."
          }
        },
        "leaf_basis": "definition_here"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "(f^{(0)})'",
        "after_tex": "f'(x)=-3+15x^2-8x^3",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [],
        "explanation": {
          "zh-Hant": "常數消失，指數逐項乘下並減一。",
          "en": "Constants vanish, while each exponent multiplies its coefficient and decreases by one."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "每個中間結果仍是多項式，下一階可微前提成立。",
            "en": "Every intermediate result remains polynomial, qualifying the next derivative."
          }
        },
        "leaf_basis": "polynomial"
      },
      {
        "id": "S03",
        "uses": [
          "S02"
        ],
        "before_tex": "(f')'",
        "after_tex": "f''(x)=30x-24x^2",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對上一階而非原函數重新微分。",
          "en": "Differentiate the preceding derivative, not the original function again."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02"
          ],
          "explanation": {
            "zh-Hant": "-3 的導數為零。",
            "en": "The derivative of -3 is zero."
          }
        },
        "leaf_basis": "polynomial"
      },
      {
        "id": "S04",
        "uses": [
          "S03"
        ],
        "before_tex": "(f'')'",
        "after_tex": "f^{(3)}(x)=30-48x",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [],
        "explanation": {
          "zh-Hant": "二階多項式在 R 上再次可微。",
          "en": "The second derivative is polynomial and differentiable again on R."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03"
          ],
          "explanation": {
            "zh-Hant": "係數 24 乘指數 2 得 48。",
            "en": "Coefficient 24 is multiplied by exponent 2, giving 48."
          }
        },
        "leaf_basis": "polynomial"
      },
      {
        "id": "S05",
        "uses": [
          "S04"
        ],
        "before_tex": "(f^{(3)})'",
        "after_tex": "f^{(4)}(x)=-48",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [],
        "explanation": {
          "zh-Hant": "四階得到常數，不可漏掉負號。",
          "en": "The fourth derivative is constant, retaining the negative sign."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S04"
          ],
          "explanation": {
            "zh-Hant": "上一階全域存在，才可談其導數。",
            "en": "The preceding derivative exists throughout the domain before differentiating it."
          }
        },
        "leaf_basis": "polynomial"
      },
      {
        "id": "S06",
        "uses": [
          "S05"
        ],
        "before_tex": "(f^{(4)})'=0",
        "after_tex": "f^{(k)}=0\\quad\\forall k\\ge5",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [],
        "explanation": {
          "zh-Hant": "五階為零；零多項式再微分仍零，以歸納涵蓋所有更高階。",
          "en": "The fifth derivative is zero; differentiating zero stays zero, so induction covers all higher orders."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S05"
          ],
          "explanation": {
            "zh-Hant": "此結論依完整多項式，而不是一般第 k 階存在推出下一階。",
            "en": "This uses the full polynomial, not a general inference from existence of one derivative to the next."
          }
        },
        "leaf_basis": "polynomial"
      },
      {
        "id": "S07",
        "uses": [
          "S03"
        ],
        "before_tex": "f''(2)=30\\cdot2-24\\cdot2^2",
        "after_tex": "f''(2)=-36",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "在階數和點確定後才代值。",
          "en": "Evaluate only after fixing the order and point."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "S03"
          ],
          "explanation": {
            "zh-Hant": "二次冪作用在 2 上，未將 24 也平方。",
            "en": "The square applies to 2, not to coefficient 24."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S02",
          "S06"
        ],
        "before_tex": "f(x)=\\sum_{j=0}^{4}c_jx^j",
        "after_tex": "f^{(k)}(x)=\\sum_{j=k}^{4}c_j\\frac{j!}{(j-k)!}x^{j-k}\\ (0\\le k\\le4)",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [
          "0\\le k\\le4"
        ],
        "explanation": {
          "zh-Hant": "連續 k 次降冪乘數是 falling factorial；k>4 已另證零。",
          "en": "After k differentiations the coefficient is the falling factorial; orders above four were separately proved zero."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "j≥k 才使用 (j-k)!，不引入負 factorial。",
            "en": "The formula uses j≥k and never a negative factorial."
          }
        },
        "leaf_basis": "polynomial"
      }
    ],
    "result": {
      "zh-Hant": "一至四階如列，五階以上皆零，二階在 2 為 -36。",
      "en": "Orders one through four are as listed, all higher orders vanish, and f''(2)=-36."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "零階導數是 0。",
          "en": "The zero-order derivative is zero."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "定義為原函數 f。",
          "en": "It is defined as f itself."
        }
      },
      {
        "claim": {
          "zh-Hant": "f''(2)=60-48。",
          "en": "f''(2)=60-48."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "第二項是 -24*2^2=-96。",
          "en": "The second term is -24*2^2=-96."
        }
      },
      {
        "claim": {
          "zh-Hant": "此 f 的第五階導數是常數 1。",
          "en": "The fifth derivative of this f is constant one."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "第四階是常數 -48，其導數為零，所有更高階也零。",
          "en": "The fourth derivative is constant -48, whose derivative is zero; every higher derivative is zero too."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "n∈N_0且每一中間階都須在宣告domain可再微分。存在第n階不推出第n+1階。多項式exact family用falling-factorial coefficients；order超過degree後為zero。高階導數存在不推出analytic或Taylor series convergence。",
          "en": "The order n belongs to N_0 and every intermediate function must be differentiable again on the declared domain. Existence of the n-th derivative does not imply existence of the next. Exact polynomial families use falling-factorial coefficients and give zero derivatives above the polynomial degree. Finite-order or even all finite-order differentiability alone does not imply analyticity or Taylor-series convergence."
        },
        "source": {
          "mko_id": "mko-higher-order-derivative",
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        }
      },
      {
        "id": "polynomial",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "多項式在實數上連續且可逐項微分；d(x^k)/dx=k x^(k-1)，常數導數為零。",
          "en": "Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero."
        }
      },
      {
        "id": "algebra",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。",
          "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities."
        }
      },
      {
        "id": "definition_here",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "零階導數是原函數；第 k+1 階是第 k 階函數在宣告域內的導數，須該導數存在。",
          "en": "The zeroth derivative is the original function; order k+1 differentiates the k-th derivative on the declared domain and requires that derivative to exist."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "second-at-two",
        "lhs": "30*2-24*2**2",
        "rhs": "-36",
        "step": "S07"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整多項式；不推一般光滑函數的解析性。",
      "en": "A full polynomial, without an analyticity claim for general smooth functions."
    },
    "md_symbols": [
      [
        "f^{(k)}",
        "第 k 階導數",
        "k-th derivative"
      ],
      [
        "c_j",
        "低到高多項式係數",
        "low-to-high coefficients"
      ]
    ],
    "md_error_controls": [
      {
        "title": "中間階 / Intermediate orders",
        "zh": "每階以前一階多項式可微為依據。",
        "en": "Each order uses differentiability of the preceding polynomial."
      },
      {
        "title": "階數零與超次數 / Zero and above-degree orders",
        "zh": "零階回 f，超過 degree 才為零。",
        "en": "Order zero returns f; only above-degree orders vanish."
      }
    ],
    "md_open_items": []
  },
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    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
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    "wording_review": "original_Lead_decisions_resolved",
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