  A widely used method for estimating Casimir interactions [H. B. G. Casimir,
Proc. K. Ned. Akad. Wet. 51, 793 (1948)] between gently curved material
surfaces at short distances is the proximity force approximation (PFA). While
this approximation is asymptotically exact at vanishing separations,
quantifying corrections to PFA has been notoriously difficult. Here we use a
derivative expansion to compute the leading curvature correction to PFA for
metals (gold) and insulators (SiO$_2$) at room temperature. We derive an
explicit expression for the amplitude $\hat\theta_1$ of the PFA correction to
the force gradient for axially symmetric surfaces. In the non-retarded limit,
the corrections to the Casimir free energy are found to scale logarithmically
with distance. For gold, $\hat\theta_1$ has an unusually large temperature
dependence.
