  Given a standard graded polynomial ring $R=k[x_1,...,x_n]$ over a field $k$
of characteristic zero and a graded $k$-subalgebra $A=k[f_1,...,f_m]\subset R$,
one relates the module $\Omega_{A/k}$ of K\"ahler $k$-differentials of $A$ to
the transposed Jacobian module $\mathcal{D}\subset \sum_{i=1}^n R dx_i$ of the
forms $f_1,...,f_m$ by means of a {\em Leibniz map} $\Omega_{A/k}\rar
\mathcal{D}$ whose kernel is the torsion of $\Omega_{A/k}$. Letting $\fp$
denote the $R$-submodule generated by the (image of the) syzygy module of
$\Omega_{A/k}$ and $\fz$ the syzygy module of $\mathcal{D}$, there is a natural
inclusion $\fp\subset \fz$ coming from the chain rule for composite
derivatives. The main goal is to give means to test when this inclusion is an
equality -- in which case one says that the forms $f_1,...,f_m$ are {\em
polarizable}. One surveys some classes of subalgebras that are generated by
polarizable forms. The problem has some curious connections with constructs of
commutative algebra, such as the Jacobian ideal, the conormal module and its
torsion, homological dimension in $R$ and syzygies, complete intersections and
Koszul algebras. Some of these connections trigger questions which have
interest in their own.
