Logarithmic Derivative of Product of Analytic Functions

Theorem
Let $D \subset \C$ be open.

Let $f, g: D \to \C$ be analytic.

Let $z \in D$ with $f \left({z}\right) \ne 0 \ne g \left({z}\right)$.

Then:
 * $\dfrac{\left({f g}\right)' \left({z}\right)} {\left({f g}\right) \left({z}\right)} = \dfrac{f' \left({z}\right)} {f \left({z}\right)} + \dfrac {g' \left({z}\right)} {g \left({z}\right)}$

Proof
Follows directly from Complex Derivative of Product.

Also see

 * Product of Analytic Functions is Analytic
 * Complex Derivative of Product
 * Logarithmic Derivative of Infinite Product of Analytic Functions