We discuss the computation of the leading corrections to D-brane solutions
due to higher derivative terms in the corresponding low energy effective
action. We develop several alternative methods for analyzing the problem. In
particular, we derive an effective one-dimensional action from which the field
equations for spherically symmetric two-block brane solutions can be derived,
show how to obtain first order equations, and discuss a few other approaches.
We integrate the equations for extremal branes and obtain the corrections in
terms of integrals of the evaluation of the higher derivative terms on the
lowest order solution. To obtain completely explicit results one would need to
know all leading higher derivative corrections which at present are not
available. One of the known higher derivative terms is the R^4 term, and we
obtain the corrections to the D3 brane solution due to this term alone. We
note, however, that (unknown at present) higher terms depending on F_5 are
expected to modify our result. We analyze the thermodynamics of brane solutions
when such quantum corrections are present. We find that the R^4 term induces a
correction to the tension and the electric potential of the D3 brane but not to
its charge, and the tension is still proportional to the electric potential
times the charge. In the near-horizon limit the corrected solution becomes
AdS_5 \times S^5 with the same cosmological constant as the lowest order
solution but a different value of the (constant) dilaton