I think the first thing I would try would be to see if the dvd drive shows up in the bios. Do you have a manual for the computer to see how to get into the bios? If not, I believe hitting F2 as the Dell logo appears when you first boot up the machine will get you into it.
The bios is used to start up and identify the components of the machine each time you boot it up. If the drive is listed in the bios it may not be entirely dead.
To get out of the bios, you’ll have to use the arrow keys to navigate and find the Exit button and hit Enter. (if you are just examining items, normally you get an option to Exit and Not Save Changes…this is a safe exit from the bios and won’t change anything in it).
Have you tried going to a restore point prior to the update?
Is the dvd drive the only part that is not working correctly? If so, and if the drive doesn’t show up in the bios, I’d call Dell and have them replace the drive.