Let $X$ be a scheme over a field $k$ and $L$ be an invertible sheaf on it. Let $D$ be the scheme over dual numbers over $k$ with parameter $t$, i.e. $Spec(k[t]/(t^2)$.

Let $X':=X \times_k D$ and $i:X \rightarrow X\times D$ the natural map.

One defines a first order deformation of $L$ over $D$ as a line bundle $L'$ on $X'$, such that $i^*L'$ is isomorphic to $L$.

One knows that the iso classes of deformations of $L$ correspond to $Ext^1_{\mathcal O_X}(\mathcal O_X, \mathcal O_X)$.

One map is clear, I think: if you have a deformation $L'$, then consider the exact sequence on $D$:

$0 \rightarrow k \rightarrow \mathcal O_D \rightarrow k \rightarrow 0$, pull it back to $X'=X\times_k D$, tensor with $L'$ and push down to $X$. Please correct me if this is not the right way.

But how do I get explicitly a deformation in the sense of the above definition out of an exact sequence on $X$

$0\rightarrow L \rightarrow M \rightarrow L \rightarrow 0$?

In the books I read there are only hints I really don't understand, so I would be very glad about an answer which carefully constructs the deformation data out of this sequence.