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

Open the communication lines with her. You can start by saying, "I noticed you behave a certain way with so-and-so. Is there anything wrong?" When you say this, remember to use a neutral tone. And when you listen to her reasons, remain neutral as well eve if they sound ridiculous to you. It could very well be that your daughter is jealous or feels that you are trying to replace her mother.