Taylor towers of symmetric and exterior powers

We study the Taylor towers of the nth symmetric and exterior power functors, Sp ⁿ and Λ ⁿ. We obtain a description of the layers of the Taylor towers, D _k Sp ⁿ and D _k Λ ⁿ, in terms of the first terms in the Taylor towers of Sp ^t and Λ ^t for t < n. The homology of these first terms is related to the stable derived functors (in the sense of Dold and Puppe) of Sp ^t and Λ ^t. We use stable derived functor calculations of Dold and Puppe to determine the lowest nontrivial homology groups for D _k Sp ⁿ and D _k Λ ⁿ.