∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {s : Set E} {ι : Type u_5} {𝔸' : Type u_7} [inst_3 : NormedCommRing 𝔸']
  [inst_4 : NormedAlgebra 𝕜 𝔸'] {u : Finset ι} {g : ι → E → 𝔸'} {g' : ι → E →L[𝕜] 𝔸'} [inst_5 : DecidableEq ι] {x : E},
  (∀ i ∈ u, HasFDerivWithinAt (g i) (g' i) s x) →
    HasFDerivWithinAt (fun x => ∏ i ∈ u, g i x) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) • g' i) s x