  We use the construction of the relative bar resolution via differential
graded structures to obtain the minimal graded free resolution of
$\text{Der}_{R \mid k}$, where $R$ is a determinantal ring defined by the
maximal minors of an $n \times (n+1)$ generic matrix and $k$ is its coefficient
field. Along the way, we compute an explicit action of the Hilbert-Burch
differential graded algebra on a differential graded module resolving the
cokernel of the Jacobian matrix whose kernel is $\text{Der}_{R \mid k}$. As a
consequence of the minimality of the resulting relative bar resolution, we get
a minimal generating set for $\text{Der}_{R \mid k}$ as an $R$-module, which,
while already known, has not been obtained via our methods.
