Theorems · Theorem · real analysis

derivWithin_fun_smul

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F]
  [inst_2 : NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} {𝕜' : Type u_2} [inst_3 : NormedRing 𝕜']
  [inst_4 : NormedAlgebra 𝕜 𝕜'] [inst_5 : Module 𝕜' F] [IsBoundedSMul 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : 𝕜 → 𝕜'},
  DifferentiableWithinAt 𝕜 c s x →
    DifferentiableWithinAt 𝕜 f s x →
      derivWithin (fun y => c y • f y) s x = c x • derivWithin f s x + derivWithin c s x • f x

Defined in

Mathlib.Analysis.Calculus.Deriv.Mul

Cited by

1 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedRingNormedAlgebraModuleIsBoundedSMulIsScalarTower