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For $\mathcal{O}$ a reduced operad, a generalized divergence from thederivations of a free $\mathcal{O}$-algebra to a suitable trace space isconstructed. In the case of the Lie operad, this corresponds to Satoh's tracemap and, for the associative operad, to the double divergence of Alekseev,Kawazumi, Kuno and Naef. The generalized divergence is shown to be a$1$-cocycle for the usual Lie algebra structure on derivations. These resultsplace the previous constructions into a unified framework; moreover, they arenatural with respect to the operad.  An important new ingredient is the use of naturality with respect to thecategory of finite-rank free modules and split monomorphisms over a commutativering $R$. This allows the notion of torsion for such functors to be exploited.  Supposing that the ring $R$ is a PID and that the operad $\mathcal{O}$ isbinary, the main result relates the kernel of the generalized divergence to thesub Lie algebra of the Lie algebra of derivations that is generated by theelements of degree one with respect to the grading induced by arity.

On derivations of free algebras over operads and the generalized divergence