Derivatives of PGF of Bernoulli Distribution/Proof 1

Proof
The Probability Generating Function of Bernoulli Distribution is:
 * $\map {\Pi_X} s = q + p s$

where $q = 1 - p$.

We have that for a given Bernoulli distribution, $p$ and $q$ are constant.

So, from Derivative of Constant, Sum Rule for Derivatives, Derivative of Identity Function and Derivative of Constant Multiple:
 * $\dfrac {\d} {\d s} \map {\Pi_X} s = p$

Again, $p$ is constant, so from Derivative of Constant:
 * $\dfrac {\d} {\d s} p = 0$

Higher derivatives are also of course zero, also from Derivative of Constant.