Theorems · Theorem · real analysis

HasFDerivWithinAt.comp_hasDerivWithinAt

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F]
  [inst_2 : NormedSpace 𝕜 F] {E : Type w} [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace 𝕜 E] {f : 𝕜 → F}
  {f' : F} (x : 𝕜) {s : Set 𝕜} {l : F → E} {l' : F →L[𝕜] E} {t : Set F},
  HasFDerivWithinAt l l' t (f x) → HasDerivWithinAt f f' s x → Set.MapsTo f s t → HasDerivWithinAt (l ∘ f) (l' f') s x

The composition l ∘ f where l : F → E and f : 𝕜 → F, has a derivative within a set equal to the Fréchet derivative of l applied to the derivative of f.

Defined in

Mathlib.Analysis.Calculus.Deriv.Comp

Cited by

7 results in Mathlib

Foundations

Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace