Did you restart the computer after the update, and after it may have automatically re-started anyway?
I'd try start up in Safe Mode, as that checked files, and also re-try the 'repair disk permissions', then
restart again to get out of Safe Mode. (Also, as separate if not first idea, try to run Disk Utility from
the Recovery mode, where OS X Utilities offers several choices, and be wary of doing more than a
'repair disk' there, since this can erase all content, among other things. It also can replace the OS X.
Safe Mode can be used to help troubleshoot or maybe even fix certain issues; generally it does no harm.
•Try safe mode if your Mac doesn't finish starting up - Apple Support
About OS X Recovery and details on how to use it:
•OS X: About OS X Recovery - Apple Support
This also can be used to restore or install a new system, so be careful to not do more than 'repair disk'
from the Disk Utility section available in OS X Utilities in recovery. This is not a permissions repair, but
that should also be one of the choices available in this mode. The more strenuous utilities available
can be used to format or erase hard drives, and other tasks, so be detail oriented & cautious.
Does your computer have a second user account? Sometimes, user account issues can be resolved
by simply logging out of the main account and running the Mac in a second user account; then restart
or move back to the default main account. (The default would be the one chosen if set to automatically
start in one user account and not require manual login.) Some users don't run an Admin as a daily user.
Anyway, troubleshooting can be hit and miss. Like some old western movie or gunsmoke re-run where
they try and try to hit cans or bottles with their shootin-iron. Except this is supposed to resolve issues
and not make broken glass or holey metal.
Good luck in any event...! 🙂