Children in the backcountry
As a father of two teenage girls and a former ski guide I have always valued getting my kids into the backcountry. They started really skiing when they were three years old, advancing to difficult slopes within a few years. Over the years they became expert skiers. Growing up skiing Alta they are into powder and steeps, so by the time they got into the BC the terrain was easily managed.
By the time they were eight years old they had skied the Baldy Chutes and Wolverine Cirque (both involve 20-45 minute booters). Since then, the extent of their backcountry experience has been limited only by motivation (they would rather rip the resort with their friends than slog about with their old man) and their lack of self-sufficiency with regards to avalanche safety. The oldest has taken her level one course and the younger takes hers next month. As your child is too young for that your best bet is to go in a group which includes at least two solid backcountry skiers or avoid potentially hazardous terrain. Look upon BC skiing with your child as you would if you were a ski guide. Everything depends on you making the right choices.
If your child has the necessary drive and equipment, and you have the experience to KNOW that a slope will be safe to ski (ie. bomber snowpack or low-angle) then you are golden. Go very (very) slow. Stop often to look around, eat snacks, stay hydrated, dig snow caves, etc. Do not demand too much self-sufficiency at first. Be prepared to carry her skis and some equipment. As she becomes proud of her exploits and abilities she will want to carry her own skis and be more self-sufficient. Only ski terrain that would be well within her abilities in a resort. Don't pick potentially hazardous terrain and make avalanche hazard an issue (be aware, but pick terrain where you won't be thinking about the possibility of digging for your child).
If you go slow and you match new challenges to her increasing interest she will enjoy the BC and cherish the experiences. One bad day when Dad pushed too hard can forestall any more days in the BC (I'm not implying you would ever push too hard, but many do).
Keep her very well hydrated and don't let her bonk. Go to the store together and shop for fuel (chocolate, fruit, energy bars, etc.). This will lead to some pre-trip anticipation and make her feel more involved.
One last thing: use tons of sunscreen.
Family values, mountain-style.
BFD, that is what its all about.