Determinant of an exact sequence sheafs

Hey everyone, I'm studying the Adjunction formula and part of the proof is confusing me. I have two exact sequences on a subvariety Y of codimension 1 in variety X:

where the are structure sheafs and (I think) are divisorial sheafs. We also have

Where these are tangent and normal bundles. The next line says "Taking determinants gives:"

which is the Adjunction formula. So can someone give me some insight into how one thinks about taking the determinants of something like an exact sequence?

Thanks!