We describe the derived functor DRep V (A) of the affine representation scheme Rep V (A) parametrizing the representations of an associative k-algebra A on a finite-dimensional vector space V. We construct the characteristic maps Tr V (A) n: HC n (A)→ H n [DRep V (A)] extending the canonical trace Tr V (A): HC 0 (A)→ k [Rep V (A)] to the higher cyclic homology of the algebra A, and describe a related derived version of the representation functor introduced recently by M. Van den Bergh. We study various operations on the homology of DRep V (A) induced by known operations on cyclic and Hochschild homology of A.