How to Fill A Deep Driveway Crack?
The shadow of the tree.
Your photo captures the shadow of the big tree next to your driveway. It is kind of artistic in that way.  You are interested in the crack but that crack is because of the tree and the presence of the tree is in the photo.  There is a big root right under the crack, lifting it up.
If you don't like the crack that root will need to go and that will take some doing.
I would like to see the tree.  There are beautiful trees in NY and I bet it is a wonder even though it has made a crack.   If you would add a picture of the tree I would appreciate it.