  We present a simple technique to compute moments of derivatives of unitary
characteristic polynomials. The first part of the technique relies on an idea
of Bump and Gamburd: it uses orthonormality of Schur functions over unitary
groups to compute matrix averages of characteristic polynomials. In order to
consider derivatives of those polynomials, we here need the added strength of
the Generalized Binomial Theorem of Okounkov and Olshanski. This result is very
natural as it provides coefficients for the Taylor expansions of Schur
functions, in terms of shifted Schur functions. The answer is finally given as
a sum over partitions of functions of the contents. One can also obtain
alternative expressions involving hypergeometric functions of matrix arguments.
