  Higher order derivative corrections to the Einstein--Maxwell action are
considered and an explicit form is found for the corrections to the entropy of
extremal black holes. We speculate on the properties of these corrections from
the point of view of small black holes and in the case when the classical black
hole potential exhibits flat directions. A particular attention is paid to the
issue of stability of several solutions, including large and small black holes
by using properties of the Hessian matrix of the effective black hole
potential. This is done by using a model independent expression for such matrix
derived within the entropy function formalism.
