If your daughter is your biological daughter (and not step-daughter or adopted), she is an American and should travel on an American passport.
ETA: There are some exceptions where you can't automatically pass American citizenship to your children, but you probably don't fit those exceptions. If you were born abroad and didn't meet a residency requirement, then she might not be American. But, if she is your daughter by birth, then she is a dual citizen, and should travel into a country she holds citizenship for on that passport to avoid complications.