The nilpotent Lie algebras dealt with in the paper under review are the naturally graded quasi-filiform Lie algebras, recently classified by J. R. Gómez and A. Jiménez-Merchán. Recall that if \({\mathfrak g}\) is a nilpotent Lie algebra, then it is quasi-filiform if \(\dim{\mathfrak g}-2\) is the least number \(k\) such that the \(k\)-th term \({\mathcal C}^k({\mathfrak g})\) of the descending central series of \({\mathfrak g}\) vanishes. On the other hand, \({\mathfrak g}\) is naturally graded if it is isomorphic to the graded Lie algebra associated to the filtration \(\{{\mathcal C}^k({\mathfrak g})\}_{k\geq 0}\). In the paper under review, the authors explicitly describe the algebra of derivations and compute the dimension of the degree-one cohomology space of \({\mathfrak g}\) with coefficients in the adjoint representation for the Lie algebras \({\mathfrak g}\) occurring in the above-mentioned classification of naturally graded quasi-filiform Lie algebras. In the final part of the paper, a program with the software \textit{Mathematica} is described, that enables one to determine the algebras of derivations for certain concrete Lie algebras.