Theorems · Theorem · real analysis

iteratedDerivWithin_scomp_two

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {g : 𝕜 → E} {f : 𝕜 → 𝕜} {s t : Set 𝕜} {x : 𝕜},
  ContDiffWithinAt 𝕜 2 g t (f x) →
    ContDiffWithinAt 𝕜 2 f s x →
      UniqueDiffOn 𝕜 t →
        UniqueDiffOn 𝕜 s →
          x ∈ s →
            Set.MapsTo f s t →
              iteratedDerivWithin 2 (g ∘ f) s x =
                derivWithin f s x ^ 2 • iteratedDerivWithin 2 g t (f x) +
                  iteratedDerivWithin 2 f s x • derivWithin g t (f x)

Defined in

Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno

Cited by

2 results in Mathlib

Foundations

Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace