How to fix a part of a washed out asphalt driveway?
That doesn't look "washed out", as in storm damage. It looks disintegrated due to age or an inadequate base. The only right fix is excavation, installation of a suitable base, and reapplication of asphalt. Anything short of that will come apart again fairly quickly. 

You wouldn't necessarily need to replace the entire driveway if this is the only troublesome area, but it looks to me like it was laid on soil that's at least partially organic.