My 10 year old daughter has temper troubles and behaves very rudely with my girlfriend. how do I find out what’s wrong?

The only way is by communicating. Or if you are on good terms with your ex-wife and she confides in her, you could probably ask for some help there. The separation from your wife itself is a loss of one parent for her already though I'm sure they get to meet regularly. Now your girlfriend seems like snatching her other parent away. Probably that's why she is defensive and unable to accept her. Go on a daddy daughter date with your girl and spend some quality time together. Shower her with love and be a gentleman to her so she feels respected. Then find another day to speak about this issue with her. You don't want her to suspect your motives in future, don't let her think that the purpose of the date is to fish out information.