  We consider the tropicalization of tangent lines to a complete intersection
curve $X$ in $\mathbb{P}^n$. 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 $\tau(X)$ of $X$. In particular, we are able to compute the
degrees of $X^*$ and $\tau(X)$ and the Newton polytope of $\tau(X)$ without
using any elimination theory.
