I have the probability generating function $G(s)$ as following. How can I get the mentioned probability-mass function $f(k)$ from it?
$$G(s)=\frac{1+\alpha(1+\mu)-\alpha(1+\mu)s}{(1+\mu-\mu s)(1+\alpha-\alpha s)}$$
$$f(k)=\Pr(X=k)=\left(1-\frac{\alpha\mu}{\mu-\alpha}\right)\frac {\mu^k}{(1+\mu)^{k+1}} + \frac {\alpha\mu} {\mu-\alpha} \frac{\alpha^k}{(1+\alpha)^{k+1}} \quad k=0,1,2,\dots $$
Note: I know that $ f(k)=\frac 1 {k!} \frac {\mathrm d^k G(s)} {\mathrm ds^k} \Big|_{s=0} $, but I do not know how it was obtained.
As the comments point out, you can certainly go backwards and show that for that $f$ you have $$G(s) = \sum_k f(k) s^k\,.$$
If you had to derive $f$ without knowing it, you can do partial fractions and use the expansion of the geometric series.