Summary: Given a representation \((\rho; A)\) of a 3-Lie algebra \(B\), we construct first-order cohomology classes by using derivations of \(A\), \(B\) and obtain a Lie algebra \(G_\rho\) with a representation \({\Phi}\) on \(H^1(B; A)\). In the case that \(\rho\) is given by an abelian extension \(0 \to A \hookrightarrow L \to B \to 0\) of 3-Lie algebras with \([A, A, L] = 0\), we obtain obstruction classes for extensibility of derivations of \(A\) and \(B\) to those of \(L\). An application of the representation \({\Phi}\) to derivations is also discussed.