# First Derivative of PGF of Negative Binomial Distribution/First Form

## Theorem

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

$\dfrac {\mathrm d} {\mathrm d s} \Pi_X \left({s}\right) = \dfrac {n p} q \left({\dfrac q {1 - ps} }\right)^{n+1}$

## Proof

$\Pi_X \left({s}\right) = \left({\dfrac q {1 - ps}}\right)^n$

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

Thus we have:

 $\displaystyle \frac {\mathrm d} {\mathrm d s} \Pi_X \left({s}\right)$ $=$ $\displaystyle \frac {\mathrm d} {\mathrm d s} \left({\dfrac q {1 - ps} }\right)^n$ $\displaystyle$ $=$ $\displaystyle n \left({\frac q {1 - ps} }\right)^{n-1} \frac {\mathrm d} {\mathrm d s} \left({\frac q {1 - ps} }\right)$ Power Rule for Derivatives and Chain Rule $\displaystyle$ $=$ $\displaystyle n \left({\frac q {1 - ps} }\right)^{n-1} \left({-p}\right) \left({\frac {-q} {\left({1 - ps}\right)^2} }\right)$ Power Rule for Derivatives and Chain Rule $\displaystyle$ $=$ $\displaystyle \frac {n p} q \left({\frac q {1 - ps} }\right)^{n+1}$ simplifying

Hence the result.

$\blacksquare$