P.S.
When she acts like that try telling her that you want her to stop now. Tell her you understand that she is frustrated because (say why) and if she doesn't stop, pick up the baby and leave the room. Keep ignoring her until she stops. The try talking to her. Children get frustrated when they feel like you don't get it and the tantrum is both an expression and a way to get what they want or get to you. With my older daughter (10) I still send her to her room, telling her that I will talk to her when she can be nice. The younger one would throw fits too and I did just what I said above. It took time and patience, but it the long run it will help you all. Remember, she is also teaching the baby how to act.
You are a good mom, otherwise you would not recognize that you need to do something different and you would not ask for help. I have also found that if you do something fun, like sing/dance with the other child, the tantrum stops pretty quickly. After all, who wants to be left out.