My wife's current legal name is SARAH SMITH. On my child's birth certificate, the line for noting the mother says
Mother's Name Prior to First Marriage which shows her name as DONGMEI HUANG.
My wife changed her name legally to SARAH SMITH during her citizenship naturalization process.
When we apply for our child's passport, how do we prove mother-child relationship?
Thanks @Traveller for your comment.
I found USCIS Form N-662 (Petition for Name Change) which shows the old name changing to the new name.