Theorems · Theorem · global analysis

ContDiffAt.deriv_fderiv_add_smul

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedAddCommGroup F]
  [inst_2 : NontriviallyNormedField 𝕜] [inst_3 : NormedSpace 𝕜 E] [inst_4 : NormedSpace 𝕜 F] {f : E → F} {x y : E}
  {t : 𝕜} {n : ℕ},
  ContDiffAt 𝕜 (↑n + 1) f (x + t • y) →
    deriv (fun s => (iteratedFDeriv 𝕜 n f (x + s • y)) fun x => y) t =
      (iteratedFDeriv 𝕜 (n + 1) f (x + t • y)) fun x => y

Defined in

Mathlib.Analysis.Calculus.TaylorIntegral

Cited by

1 results in Mathlib

Foundations

Depth 188 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupNontriviallyNormedFieldNormedSpaceNormedSpace