Theorems · Theorem · real analysis

iteratedDerivWithin_vcomp_three

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {g : E → F} {f : 𝕜 → E}
  {s : Set 𝕜} {t : Set E} {x : 𝕜},
  ContDiffWithinAt 𝕜 3 g t (f x) →
    ContDiffWithinAt 𝕜 3 f s x →
      UniqueDiffOn 𝕜 t →
        UniqueDiffOn 𝕜 s →
          x ∈ s →
            Set.MapsTo f s t →
              iteratedDerivWithin 3 (g ∘ f) s x =
                ((iteratedFDerivWithin 𝕜 3 g t (f x)) fun x_1 => derivWithin f s x) +
                      (iteratedFDerivWithin 𝕜 2 g t (f x)) ![iteratedDerivWithin 2 f s x, derivWithin f s x] +
                    2 • (iteratedFDerivWithin 𝕜 2 g t (f x)) ![derivWithin f s x, iteratedDerivWithin 2 f s x] +
                  (fderivWithin 𝕜 g t (f x)) (iteratedDerivWithin 3 f s x)

Defined in

Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno

Cited by

2 results in Mathlib

Foundations

Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace