  In this paper, we study convergence and superconvergence theory of integer
and fractional derivatives of the one-point and the two-point Hermite
interpolations. When considering the integer-order derivative, exponential
decay of the error is proved, and superconvergence points are located, at which
the convergence rates are $O(N^{-2})$ and $O(N^{-1.5})$, respectively, better
than the global rate for the one-point and two-point interpolations. Here $N$
represents the degree of interpolation polynomial. It is proved that the
$\alpha$-th fractional derivative of $(u-u_N)$ with $k<\alpha<k+1$, is bounded
by its $(k+1)$-th derivative. Furthermore, the corresponding superconvergence
points are predicted for fractional derivatives, and an eigenvalue method is
proposed to calculate the superconvergence points for the Riemann-Liouville
fractional derivative. In the application of the knowledge of superconvergence
points to solve FDEs, we discover that a modified collocation method makes
numerical solutions much more accurate than the traditional collocation method.
