  The exceptional zero phenomenon has been widely studied in the realm of
$p$-adic $L$-functions, where the starting point lies in the foundational work
of Mazur, Tate and Teitelbaum. This phenomenon also appears in the study of
Euler systems, which comes as no surprise given the interaction between these
two settings. When this occurs, one is led to study higher order derivatives of
the Euler system in order to extract the arithmetic information which is
usually encoded in the explicit reciprocity laws. In this work, we focus on the
elliptic units of an imaginary quadratic field and study this exceptional zero
phenomenon, proving an explicit formula relating the logarithm of a {\it
derived} elliptic unit either to special values of Katz's two variable $p$-adic
$L$-function or to its derivatives. Further, we interpret this fact in terms of
an $\mathcal L$-invariant, and relate this result to other approaches to the
exceptional zero phenomenon concerning Heegner points and Beilinson--Flach
elements.
