Theorems · Theorem · real analysis

logDeriv_prod

∀ {𝕜 : Type u_1} {𝕜' : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NontriviallyNormedField 𝕜']
  [inst_2 : NormedAlgebra 𝕜 𝕜'] {ι : Type u_3} {s : Finset ι} {f : ι → 𝕜 → 𝕜'} {x : 𝕜},
  (∀ i ∈ s, f i x ≠ 0) →
    (∀ i ∈ s, DifferentiableAt 𝕜 (f i) x) → logDeriv (fun x => ∏ i ∈ s, f i x) x = ∑ i ∈ s, logDeriv (f i) x

The logarithmic derivative of a finite product is the sum of the logarithmic derivatives.

Defined in

Mathlib.Analysis.Calculus.LogDeriv

Cited by

4 results in Mathlib

Foundations

Depth 190 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNontriviallyNormedFieldNormedAlgebra