It all depends on the functions your application will need to complete.
If you will have several functions that edit, several functions that delete, and several functions that update, all functions having separate needs and requirements (thus cannot be made into a single universal function), then separate files for each function (edit_ajax.php, update_ajax.php, etcetera) would be easiest in the long run.
Or, are these Ajax calls only a few of the ones you'll need involving Ajax? In that case, it may be better to group them by their domain function, which is what they're handling (students_ajax.php if those Ajax calls involve updating, deleting and editing students, for example).
Also, make sure your function calls are separated by concern if your application architecture supports it. If these calls are purely sending data to PHP to update your database, keep them in a separate file from the Ajax calls that are merely passing parameters in for a specific set of HTML to load into a page.
Ultimately, your choice is heavily dependent on your application architecture and design style, which may sound more obvious than needed but very true nonetheless.