$\frac{\partial}{\partial t} e^{\lambda e^{it-1}}$
$\frac{\partial}{\partial t}\left( 1-\frac{it}{\alpha}\right)^{-\beta}$
I calculated the characteristic functions of the Poisson and Gamma distributions to be the above function and am trying to calculate expectation and variance from them.
It has been years since I took calculus and I am pretty unsure how to take the derivatives.
I do know that I can find $E(X)$ by taking the first derivative of the characteristic function and evaluating it at t=0, (and similar for higher moments) and I know what the answers for expectation and variance are for these distributions, but I am struggling with the calculus.
Any help is appreciated