  We investigate interconnected aspects of hyperderivatives of polynomials over
finite fields, q-th powers of polynomials, and specializations of Vandermonde
matrices. We construct formulas for Carlitz multiplication coefficients using
hyperderivatives and symmetric polynomials, and we prove identities for
hyperderivative power sums in terms of specializations of the inverse of the
Vandermonde matrix. As an application of these results we give a new proof of a
theorem of Thakur on explicit formulas for Anderson's special polynomials for
log-algebraicity on the Carlitz module. Furthermore, by combining results of
Pellarin and Perkins with these techniques, we obtain a new proof of Anderson's
theorem in the general case.
