  In this article we study the differentiability of solutions of parabolic
semilinear stochastic evolution equations (SEEs) with respect to their initial
values. We prove that if the nonlinear drift coefficients and the nonlinear
diffusion coefficients of the considered SEEs are $n$-times continuously
Fr\'{e}chet differentiable, then the solutions of the considered SEEs are also
$n$-times continuously Fr\'{e}chet differentiable with respect to their initial
values. Moreover, a key contribution of this work is to establish suitable
enhanced regularity properties of the derivative processes of the considered
SEE in the sense that the dominating linear operator appearing in the SEE
smoothes the higher order derivative processes.
