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We study the Taylor towers of the nth symmetric and exterior power functors, Spⁿ and Λⁿ. We obtain a description of the layers of the Taylor towers, $D_{k}Spⁿ$ and $D_k\Lambda ⁿ$, in terms of the first terms in the Taylor towers of $S{p}^t$ and ${\Lambda }^t$ for t < n. The homology of these first terms is related to the stable derived functors (in the sense of Dold and Puppe) of $S{p}^t$ and ${\Lambda }^t$. We use stable derived functor calculations of Dold and Puppe to determine the lowest nontrivial homology groups for $D_{k}Spⁿ$ and $D_k\Lambda ⁿ$.