  Generalising a previous work of Jiang and Sheng, a cohomology theory for
differential Lie algebras of arbitrary weight is introduced. The underlying
$L_\infty[1]$-structure on the cochain complex is also determined via a
generalised version of higher derived brackets. The equivalence between
$L_\infty[1]$-structures for absolute and relative differential Lie algebras
are established. Formal deformations and abelian extensions are interpreted by
using lower degree cohomology groups. Also we introduce the homotopy
differential Lie algebras. In a forthcoming paper, we will show that the operad
of homotopy (relative) differential Lie algebras is the minimal model of the
operad of (relative) differential Lie algebras.
