  We show that, in the model constructed by adding sufficiently many Cohen
reals, derived limits are additive on a large class of systems. This
generalizes the work of Jeffrey Bergfalk, Michael Hru\v s\'ak, and Chris
Lambie-Hanson which focuses on the system $\mathbf{A}$. In the process, we
isolate a partition principle responsible for the vanishing of derived limits
on collections of Cohen reals and reframe the propagating trivializations
result of as a theorem of ZFC. In light of results of the author, Jeffrey
Bergfalk, and Justin Moore in \cite{ASH}, the additivity of derived limits also
implies additivity results for strong homology.
