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

## Theorem

Then the first derivative of the PGF of $X$ with respect to $s$ 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

$\displaystyle \map {\Pi_X} s = \paren {\frac {p s} {1 - q s} }^n$

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

Thus we have:

 $\ds \frac \d {\d s} \map {\Pi_X} s$ $=$ $\ds \frac \d {\d s} \paren {\frac {p s} {1 - q s} }^n$ $\ds$ $=$ $\ds n \paren {\frac {p s} {1 - q s} }^{n - 1} \frac \d {\d s} \paren {\frac {p s} {1 - q s} }$ Power Rule for Derivatives and Chain Rule for Derivatives $\ds$ $=$ $\ds n \paren {\frac {p s} {1 - q s} }^{n - 1} \paren {\frac {\paren {1 - q s} \frac \d {\d s} \paren {p s} - p s \frac \d {\d s} \paren {1 - q s} } {\paren {1 - q s}^2} }$ Quotient Rule for Derivatives and Chain Rule for Derivatives $\ds$ $=$ $\ds n \paren {\frac {p s} {1 - q s} }^{n - 1} \paren {\frac {\paren {1 - q s} p - p s \paren {-q} } {\paren {1 - q s}^2} }$ Power Rule for Derivatives where $n = 1$ $\ds$ $=$ $\ds n \paren {\frac {p s} {1 - q s} }^{n - 1} \paren {\frac {p - p q s + p q s} {\paren {1 - q s}^2} }$ $\ds$ $=$ $\ds n \paren {\frac {p s} {1 - q s} }^{n - 1} \paren {\frac p {\paren {1 - q s}^2} }$ $\ds$ $=$ $\ds n p \paren {\frac {\paren {p s}^{n - 1} } {\paren {1 - q s}^{n + 1} } }$

Hence the result.

$\blacksquare$