Summary: We describe the derivations of a direct limit of Lie superalgebras \(\mathfrak L^i\) \((i \in I)\) in an \(\mathfrak L\)-module \(\mathfrak u\) as the inverse limit of the derivations of \(\mathfrak L^i\)’s in \(\mathfrak u\). Using this, in case the first cohomology group of each \(\mathfrak L^i\) with coefficients in \(\mathfrak u\) is zero, we describe the derivations of \(\mathfrak L\) in \(\mathfrak u\) as the inverse limit of \(\mathfrak u/\mathfrak u^{\mathfrak L^i}\) \((i\in I)\). This then allows us to compute the derivations of direct limits of finite-dimensional basic classical simple Lie superalgebras.