  We define bivariant algebraic K-theory and bivariant derived Chow on the
homotopy category of derived schemes over a smooth base. The orientation on the
latter corresponds to virtual Gysin homomorphisms. We then provide a morphism
between these two bivariant theories and compare the two orientations. This
comparison then yields a homological and cohomological
Grothendieck-Riemann-Roch formula for virtual classes.
