Theorems · Theorem · real analysis

ContinuousLinearMap.derivWithin_of_bilinear

∀ {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] {G : Type u_1}
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {x : 𝕜} {s : Set 𝕜} {B : E →L[𝕜] F →L[𝕜] G} {u : 𝕜 → E}
  {v : 𝕜 → F},
  DifferentiableWithinAt 𝕜 u s x →
    DifferentiableWithinAt 𝕜 v s x →
      derivWithin (fun y => (B (u y)) (v y)) s x = (B (u x)) (derivWithin v s x) + (B (derivWithin u s x)) (v x)

Defined in

Mathlib.Analysis.Calculus.Deriv.Mul

Cited by

0 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace