Theorems · Theorem · real analysis

iteratedDeriv_vcomp_two

∀ {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} {x : 𝕜},
  ContDiffAt 𝕜 2 g (f x) →
    ContDiffAt 𝕜 2 f x →
      iteratedDeriv 2 (g ∘ f) x =
        ((iteratedFDeriv 𝕜 2 g (f x)) fun x_1 => deriv f x) + (fderiv 𝕜 g (f x)) (iteratedDeriv 2 f x)

Defined in

Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno

Cited by

0 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace