  The objective of this paper is to clarify the relationship between the
$C^1$-smooth dependence of solutions to delay differential equations (DDEs) on
initial histories (i.e., initial conditions) and delay parameters. For this
purpose, we consider a class of DDEs which include a constant discrete delay.
The problem of $C^1$-smooth dependence is fundamental from the viewpoint of the
theory of differential equations. However, the above mentioned relationship is
not obvious because the corresponding functional differential equations have
the less regularity with respect to the delay parameter. In this paper, we
prove that the $C^1$-smooth dependence on initial histories and delay holds by
adopting spaces of initial histories of Sobolev type, where the
differentiability of translation in $L^p$ plays an important role.
