Derivative of Generating Function for Sequence of Harmonic Numbers

Theorem
Let $\sequence {a_n}$ be the sequence defined as:
 * $\forall n \in \N_{> 0}: a_n = H_n$

where $H_n$ denotes the $n$th harmonic number.

Let $\map G z$ be the generating function for $\sequence {a_n}$:
 * $\map G z = \dfrac 1 {1 - z} \map \ln {\dfrac 1 {1 - z} }$

from Generating Function for Sequence of Harmonic Numbers.

Then the derivative of $\map G z$ $z$ is given by:


 * $\map {G'} z = \dfrac 1 {\paren {1 - z}^2} \map \ln {\dfrac 1 {1 - z} } + \dfrac 1 {\paren {1 - z}^2}$