zpow — Mathlib

English

Again, the k-th derivative of the reciprocal function is given by (-1)^k k! x^{-(1+k)}.

Русский

Снова k-я производная обратной функции равна (-1)^k k! x^{-(1+k)}.

LaTeX

$$$\\mathrm{deriv}^[k] (x \\mapsto x^{-1}) = (-1)^k k! \\cdot x^{-(1+k)}$$$

Lean4

theorem zpow (hf : DifferentiableWithinAt 𝕜 f t a) (h : f a ≠ 0 ∨ 0 ≤ m) :
    DifferentiableWithinAt 𝕜 (fun x => f x ^ m) t a :=
  (differentiableAt_zpow.2 h).comp_differentiableWithinAt a hf

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