First Derivative of PGF of Negative Binomial Distribution/First Form

Theorem
Let $X$ be a discrete random variable with the negative binomial distribution (first form) with parameters $n$ and $p$.

Then the first derivative of the PGF of $X$ w.r.t. $s$ is:


 * $\dfrac \d {\d s} \map {\Pi_X} s = \dfrac {n p} q \paren {\dfrac q {1 - p s} }^{n + 1}$

Proof
The Probability Generating Function of Negative Binomial Distribution (First Form) is:


 * $\map {\Pi_X} s = \paren {\dfrac q {1 - p s} }^n$

We have that for a given negative binomial distribution, $n, p$ and $q$ are constant.

Thus we have:

Hence the result.