The USA does not recognise dual nationalities. So yes, you must have your US passport. So if you are a dual national, you need to bring both passports with you. The US passport for entering the US, and your other passport for coming back. It's no different for children. I still remember they effing FINGER PRINTED my young son at US immigration when we visited stateside when he was only around the age of 4. Couldn't believe that. Even now, even though I was born in California, because I've lived here so long, I have to disclose where I'm staying when I visit stateside. I find that quite intrusive and out of order as again I was born in California, but that's the way they are. And ironic given many immigration officers look hispanic or speak as if they are not native to the USA. A bit rich, right? Some official asking you what your intent is when you are from there, and they aren't. You have to laugh.
And keep in mind it's an absolute pain in the **** to renew US passports over here. You either have to trek to London (OK if you live in London but not OK if you dont), and pay for a hotel given the latest embassy appointments are around 10am, or you have to use their useless courier, DX, who *bleep* up everything habitually.