My nine-year-old daughter is becoming too dependent on her mother for every small thing. How do I teach her to be independent?

Getting her to do the tasks herself and speaking to your wife to not help your daughter every time she asks for your wife's assistance. Also, try assigning her simple tasks around the house such as clearing the tables, helping with folding of laundry etc and praising her when she does the tasks well may help. Worked for one of my friends. :)