The mouthing off and disrespect is pretty normal at this stage. She is testing to see what behavior is acceptable and what is not. Have a talk with her when everyone is calm. Explain in a matter of fact way that back talk and disrespect is not acceptable. Then tell her exactly what type of behavior is expected. Not because you're the boss, but because that is the right way for people to treat each other. She needs to learn that there is a specific consequence for a specific behavior. Explain the consequences to her. Then tell her exactly what will happen when she behaves inappropriately. My son (who is five) really likes playing video games on the weekend, so losing that privilege works. He gets a warning when is behaving in an unacceptable way. If he doesn't stop, he gets a mark on the calendar. If he gets five marks in a week, he doesn't get to play video games on the weekend. You might need a more immediate consequence. You'll have to decide what consequences work best in your situation. It's just as important to acknowledge good behavior. I make a point of telling him is doing a good job, and I thank him for his good behavior. Consistency is the key. You must follow through every time. When you don't, she will get confused and frustrated, and not learn for sure what the rules are. It would best if your husband would come on board. But if not, just make sure you are consistent.
Watch your own behavior carefully and make sure you are not being disrespectful to her. If she needs your attention, acknowledge her request right away. I don’t mean jump when she says jump. If you can’t, or don’t want to do what she is asking, at least acknowledge her need right away, and then explain why you cannot do what she wants at the moment. Don’t get into an argument even if she tries. Tell her you have already explained it to her, and then quit talking. My son will argue until the cows come home, but only if I am a willing participant. If I don’t argue, then he is just talking to himself. He eventually gives up, and learns that badgering me doesn’t work.
Sometimes we get so busy that we don’t realize we are treating our kids disrespectfully by not being attentive to their requests. I know I can fall into that pattern when I’m in a hurry. I sometimes forget my son is just a little guy and needs more time to get ready, or gets tired and hungry more often than I do. I constantly try and check myself when a situation is getting out of hand to see how I might be adding to his frustration, which leads to his contrary behavior.
Don’t take any of it personally, or feel like your daughter is headed toward delinquency. Kids are always in a state of trying to learn the rules of life. It is up to us as parents to stay calm, not get emotionally thrown off balance, and just keep teaching.
PS-My son acts totally different at school and at grandma's. It seems to be me he is testing most of the time. I am the one who spends the most time with him, so that makes sense.