I am sorry to hear this. There is a 10.10 compatible driver for this Ethernet adpater available -> http://plugable.com/products/USB3-E10….
*BUT* it is best practice to uninstall drivers before performing a major upgrade and to install the newest version after the upgrade.
There have also been cases where customers complained about not having any network functionality after the upgrade which is a different problem all together. Please test to see if your WiFi connection works. If so, we can rule out this problem.
Then I would recommend the following (please make sure to read this carefully, you do not want to accidentally delete the wrong extension or make a mistake when performing these steps, I cannot take responsibility for things going wrong here):
Disconnect Ethernet adapter form computer
Take a look at System Information > Software > Extensions and look for an instance labeled “AX88179_798A” and select it and take a look at the “Location path” (as an example, in Mavericks it was /Library/Extensions/AX88179_178A.kext)
Open your Terminal and run the following command -> sudo kextunload /path ofwherever/thextensionis/NAME_OF_THE_KEXT_FILE.kext (note this is the path shown in system information in step 1). Now you have unloaded this extension.
Download the newest driver from here -> http://plugable.com/products/USB3-E10… and install it
Connect Ethernet adapter and test
Please let me know if this yields any results.