I'm trying to do point (b) of exercise 3.30 of the book "Statistical Inference" (Casella & Berger).
The exercise says:
"Use the identities of Theorem 3.4.2. to
(a) calculate the variance of a binomial random variable.
(b) calculate the mean and variance of a beta(a,b) random variable."
Theorem 3.4.2 says the following.
If X is a random variable with pdf or pmf of the form
(exponential family), then
and .

Point (a) is doable, but what about point (b)? First of all, since we have two parameters (a and b), do we get a system of two equations? My difficulties is that I have a cumbersome E[logX] and then I have to take derivatives of B(a,b). It's a nightmare... If I'm not wrong, .
Any help?

In my Casella and Berger, part b is asking for instead of Beta distribution. Not sure if this method is useful for Beta distribution. Anyway you may take a look at

and there are lots of calculations for and other related terms. And as you expected you will need to in terms of some non-elementary function like the digamma function which is the derivative of log-gamma function.