iteratedDeriv_scomp_eq_sum_orderedFinpartition

English

A Faà di Bruno-type identity for the third derivative of a composition within sets, with vector-valued outputs.

Русский

Идентичность Фаàди Бруно для третьей производной композиции внутри множеств векторного значения.

LaTeX

$$$\\forall {g,f}{s,t}{x},\\ ContDiffWithinAt 𝕜 3 g t (f x) \\to ContDiffWithinAt 𝕜 3 f s x \\Rightarrow \\n Eq (iteratedDerivWithin 3 (Function.comp g f) s x) (\\text{сложная сумма по разбиениям})$$$

Lean4

theorem iteratedDeriv_scomp_eq_sum_orderedFinpartition (hg : ContDiffAt 𝕜 n g (f x)) (hf : ContDiffAt 𝕜 n f x)
    (hi : i ≤ n) :
    iteratedDeriv i (g ∘ f) x =
      ∑ c : OrderedFinpartition i, (∏ j, iteratedDeriv (c.partSize j) f x) • iteratedDeriv c.length g (f x) :=
  by
  rw [iteratedDeriv_vcomp_eq_sum_orderedFinpartition hg hf hi]
  simp only [iteratedFDeriv_apply_eq_iteratedDeriv_mul_prod]

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