Derivative of the Riemann-Hilbert map

Given a pair $(X,\mathrm{\nabla })$, consisting of a closed Riemann surface $X$ and a holomorphic connection $\mathrm{\nabla }$ on the trivial principal bundle $X\times {\mathrm{S}\mathrm{L}}_2(\mathbb{C})\to X$, the Riemann-Hilbert map sends $(X,\mathrm{\nabla })$ 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.