  The $j$-invariant of a cubic curve is an isomorphism invariant parameterized
by the moduli space of elliptic curves. The Hesse derivative of a curve $V(f)$
given by the homogeneous polynomial $f$ is $V(\mathcal{H}(f))$ where
$\mathcal{H}(f)$ is a the determinant of the Hesse matrix of $f$. In this
paper, we compute the $j$-invariant of the Hesse derivative of a cubic curve
$C$ in terms of the $j$-invariant of $C$, getting a rational function on the
Riemann sphere. We then analyze the dynamics of this rational function, and
investigate when a cubic curve is isomorphic to its $n$-fold Hesse derivative.
