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Stopping NetBeans also stops the Derby Server. When NetBeans is restarted, you can still find in the Runtime tab the same connections that were present before NetBeans was stopped. Since it's obvious that these connections no longer work, why are they not simply cleaned away?
Right click on the connection. The options are "Connect", "Properties", and "Delete". The Properties window allows you to change the Database URL and User. This is very convenient if your database has multiple schemas and users. The state of the connection (Closed and Connected) is reflected in the node. It's a convenience to have the connection status graphically displayed. Users can delete the connection node if they so choose. I would vote against fixing this "defect" if that were possible.
This is not a defect, it is how it is supposed to work. After restart you can connect the connections (perhpas to another schema, or using another username) and the Derby server will be started for you.