For passport you use form DS11 it has instructions on the second page for what proof you need for your child
https://eforms.state.gov/Forms/ds11.pdf
APPLICANTS BORN OUTSIDE THE UNITED STATES:
- If you claim citizenship through naturalization of parent(s): Submit the Certificate(s) of Naturalization of your parent(s), your foreign birth certificate (and official translation if the document is not in English), proof of your admission to the United States for permanent residence, and your parents' marriage/certificate and/or evidence that you were in the legal and physical custody of your U.S. citizen parent, if applicable.
also note from page 1
REQUIREMENTS FOR CHILDREN AS DIRECTED BY PUBLIC LAW 106-113 AND 22 CFR 51.28:
To submit an application for a child under age 16 both parents or the child's legal guardian(s) must appear and present the following:
- Evidence of the child's U.S. citizenship;
- Evidence of the child's relationship to parents/guardian(s); AND
- Original parental/guardian government-issued identification AND a photocopy of the front and back side of presented identification.
IF ONLY ONE PARENT APPEARS, YOU MUST ALSO SUBMIT ONE OF THE FOLLOWING:
- Second parent's notarized written statement or DS-3053 (including the child's full name and date of birth) consenting to the passport issuance for the child. The notarized statement cannot be more than three months old and must be signed and notarized on the same day, and must come with a photocopy of the front and back side of the second parent's government-issued photo identification; OR
- Second parent’s death certificate if second parent is deceased; OR
- Primary evidence of sole authority to apply, such as a court order; OR
- A written statement or DS-5525 (made under penalty of perjury) explaining in detail the second parent's unavailability.
AS DIRECTED BY REGULATION 22 C.F.R. 51.21 AND 51.28:
- Each minor child applying for a U.S. passport book and/or passport card must appear in person.
