  Given a pair $(X,\nabla)$, consisting of a closed Riemann surface $X$ and a
holomorphic connection $\nabla$ on the trivial principal bundle
$X\times\mathrm{SL}_2(\mathbb{C})\to X$, the Riemann-Hilbert map sends
$(X,\nabla)$ to its monodromy representation. We compute the derivative of this
map, and provide a simple description of the locus where it is injective,
recovering in the process several previously obtained results.
