First Derivative of PGF of Negative Binomial Distribution/Second Form

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

Then the first derivative of the PGF of $X$ $s$ is:


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

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


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

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

Thus we have:

Hence the result.