Let τ be an action of a compact Lie groupGon an unital C*‐algebraA, and letAFbe theG‐finite elements inA. We show that if τ satisfies a certain spectral condition, then any*‐derivation δ defined onAF, and mappingAF, is closable and its closure generates a one‐parameter group of*‐automorphisms. The condition on τ is fulfilled in the cases where τ has full spectrum, or ifAseparating family ï ∈Ĝis associated with Hilbert spaces inAin the sense of Roberts, or ifGis abelian and the condition Γ of [7] is fulfilled. The proof is based on an argument of K. Thomsen, who proved the theorem in the case whereGis abelian [25]. A more detailed study is made of the cononical action ofG=U(n)on the Cuntz's algebraCn, and, in particular, we find that any derivation onAFvanshing on the fixed point algebra for this action is the generator of a one‐parameter subgroup fo the action.