  We construct canonical strongly bicovariant differential graded algebra
structures on all four flavours of cross product Hopf algebras, namely double
cross products $A\hookrightarrow A\bowtie H\hookleftarrow H$, double cross
coproducts $A\twoheadleftarrow A {\blacktriangleright\!\!\blacktriangleleft}
H\twoheadrightarrow H$, biproducts $A{\buildrel\hookrightarrow\over
\twoheadleftarrow}A{\cdot\kern-.33em\triangleright\!\!\!<} B$ and
bicrossproducts $A\hookrightarrow A{\blacktriangleright\!\!\triangleleft}
H\twoheadrightarrow H$ on the assumption that the factors have strongly
bicovariant calculi $\Omega(A),\Omega(H)$ (or a braided version $\Omega(B)$).
We use super versions of each of the constructions. Moreover, the latter three
quantum groups all coact canonically on one of their factors and we show that
this coaction is differentiable. In the case of the Drinfeld double
$D(A,H)=A^{\rm op}\bowtie H$ (where $A$ is dually paired to $H$), we show that
its canonical actions on $A,H$ are differentiable. Examples include are a
canonical $\Omega(\Bbb C_q[GL_2\ltimes \Bbb C^2])$ for the quantum group of
affine transformations of the quantum plane and $\Omega(\Bbb C_\lambda[{\rm
Poinc_{1,1}}])$ for the bicrossproduct Poincar\'e quantum group in 2
dimensions. We also show that $\Omega(\Bbb C_q[GL_2])$ itself is uniquely
determined by differentiability of the canonical coaction on the quantum plane
and of the determinant subalgebra.
