∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {x : 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'} {f' : ι → 𝔸'},
  (∀ i ∈ u, HasStrictDerivAt (f i) (f' i) x) →
    HasStrictDerivAt (fun x => ∏ i ∈ u, f i x) (∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • f' i) x