  We develop an effective algorithm to compute the derivative of a Bianchi
modular form with respect to weight space as it varies in a $p$-adic family.
This method is entirely local at the modular form, and does not compute the
family anywhere outside an infinitesimal neighbourhood. We numerically verify
some conjectures surrouding smoothness of the eigenvariety (equivalently,
uniqueness of families) and the ``direction over weight space'' of the family.
The methods are also applied to study elliptic modular forms and their
$\mathcal{L}$-invariants.
