Theorems · Theorem · real analysis

derivWithin_fun_finsetProd

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} {ι : Type u_2} [inst_1 : DecidableEq ι]
  {𝔸' : Type u_3} [inst_2 : NormedCommRing 𝔸'] [inst_3 : NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {f : ι → 𝕜 → 𝔸'},
  (∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x) →
    derivWithin (fun x => ∏ i ∈ u, f i x) s x = ∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • derivWithin (f i) s x

Defined in

Mathlib.Analysis.Calculus.Deriv.Mul

Cited by

2 results in Mathlib

Foundations

Depth 189 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldDecidableEqNormedCommRingNormedAlgebra