I too have been dealing with touchpad randomly shutting off on Dell Insprion laptop with windows 10. Searching through device manager and mouse settings showed nothing for touchpad, and no dirvers found on Dell Support site either.
Instead what I discovered after much searching, is that there is a setting for turning touchpad on/off hidden in the windows settings. Open up Settings in the Start menu, then search for Windows Mobility Center. Click to open. You will find settings for both Touchpad on/off and Function key behaviour there.
I have read of some laptops with Fn keys that toggle touchpad settings. Hence I've changed my laptop so Fn keys behave as function keys, not multimedia.