On a day dominated by news that Mr. Trump had finally dropped his long adherence to the false notion that President Obama was not born in the United States — and therefore is not a legitimate president — Mrs. Obama said her husband had faced a long list of doubts since his election in 2008 as the first black president.

“Back then, people had all kinds of questions about what kind of president Barack would be,” she said. “Things like, does he understand us, will he protect us, and then, of course, there were those who questioned — and continue to question for the past eight years, up through this very day — whether my husband was even born in this country.”

Boos echoed through the hall as Mrs. Obama pursed her lips in disdain.

Mrs. Obama cast Mrs. Clinton — herself a former first lady and Mr. Obama’s onetime rival — as the only candidate capable of fulfilling what she called “the highest-stakes, most 24/7 job that you could imagine.”

“With every word that they utter, they can start wars, crash markets, change the course of this planet, so who in this election is truly ready for this job?” Mrs. Obama said. “It is excruciatingly clear that there is only one person in this election we can trust with those responsibilities, only one person with the qualifications and the temperament for that job, and that is our friend Hillary Clinton.”