iteratedDeriv_add — Mathlib

English

If f is ContDiffOn, then the nth derivative of f composed with negation x ↦ f (−x) satisfies a sign pattern, namely (−1)^n times the nth derivative of f evaluated at (−x).

Русский

Если f ∈ ContDiffOn, то n-я производная композиции f(−x) имеет знак (−1)^n, то есть равна (−1)^n умножить на n-ю производную f(-x).

LaTeX

$$$\\operatorname{iteratedDeriv}_n (\\lambda x. f (-x)) a = (-1)^n \\cdot \\operatorname{iteratedDeriv}_n f (-a)$$$

Lean4

theorem iteratedDeriv_add (hf : ContDiffAt 𝕜 n f x) (hg : ContDiffAt 𝕜 n g x) :
    iteratedDeriv n (f + g) x = iteratedDeriv n f x + iteratedDeriv n g x := by
  simpa only [iteratedDerivWithin_univ] using iteratedDerivWithin_add (Set.mem_univ _) uniqueDiffOn_univ hf hg

Похожие утверждения