Let us consider $0\to E\to G\to H\to 0$ an exact sequence of coherent sheaves on a surface $S$ such that $rk G=rk E +1$ and the double dual of $H$ is $\mathcal{O}_S(D)$ for some effective divisor $D$. Is  it true that I have the following exact sequence:

$0\to detE\to det G\to i_*\mathcal{O}_D\to 0$,

where $i$ is the inclusion map from $D$ to $S$?
In general, which is the behaviour of the determinat with respect to a short exact sequence of sheaves?