  We prove that the tangent complex of K-theory, in terms of (abelian)
deformation problems over a characteristic 0 field k, is cyclic homology (over
k). This equivalence is compatible with the $\lambda$-operations. In
particular, the relative algebraic K-theory functor fully determines the
absolute cyclic homology over any field k of characteristic 0.
  We also show that the Loday-Quillen-Tsygan generalized trace comes as the
tangent morphism of the canonical map $BGL_\infty \to K$.
  The proof builds on results of Goodwillie, using Wodzicki's excision for
cyclic homology and formal deformation theory \`a la Lurie-Pridham.
