I'm assuming you have a keyless chuck. Turn it to its highest setting and tighten the chuck. I usually have trouble with drilling (forwards) and the chuck coming loose. I've found if I turn my drills to the highest clutch setting then tighten the chuck it then doesn't come loose.

Thanks, but I'm still confused. It seems like no matter how tight you tighten the chuck, once you put the drill in reverse, putting the bit into a well-set screw has the same effect as holding the chuck with your hand, once you squeeze the trigger, the chuck loosens up. I'm assuming there's no way around this?

This wasn't a problem until I fell and hurt my wrist, now using a screwdriver doesn't work so well. But, I'm thinking you really need a screwdriver to "break" the screw loose, then you can "spin" it the rest of the way out with the drill. But I'm still hoping I'm missing something.

It shouldnt do this,I tighten mine on torq setting not speed setting,slower more torq will set it better.If yoyu a faster speed on the drill (no not the # 15) most will have a switch that is A or B or 1 and 2 this is setting for fast screwing and slower but more torq of course you must also have the driver set on the highest # possible.
If still not working have someone with a great grip give it a try or get one that works for you I break screws loose all the time and so do the contractors I paint for,otherwise we'd all be out of business.

Thanks!

There is a problem with your chuck assembly. If this is a dewalt drill, 1/2 inch, you can repalce this with a ratcheting chuck assembly, and stop the proablem. If it is 3/8 x21 thread, you can put a keyed chuck, or keyless. I feel you have a defective chuck assembly, one of the dogs inside the chuck assembly is not holding.