  We study the smoothness of the solution of the directed chain stochastic
differential equations, where each process is affected by its neighborhood
process in an infinite directed chain graph, introduced by Detering et al.
(2020). Because of the auxiliary process in the chain-like structure, classic
methods of Malliavin derivatives are not directly applicable. Namely, we cannot
make a connection between the Malliavin derivative and the first order
derivative of the state process. It turns out that the partial Malliavin
derivatives can be used here to fix this problem.
