Tropical tangents for complete intersection curves

We consider the tropicalization of tangent lines to a complete intersection curve X in ℙⁿ. Under mild hypotheses, we describe a procedure for computing the tropicalization of the image of the Gauss map of X in terms of the tropicalizations of the hypersurfaces cutting out X. We apply this to obtain descriptions of the tropicalization of the dual variety X^∗ and tangential variety τ(X) of X. In particular, we are able to compute the degrees of X^∗and τ(X) and the Newton polytope of τ(X) without using any elimination theory.