Donald Trump speaks during the Republican presidential debate at the Reagan library on Sept. 16, 2015, in Simi Valley, Calif.(Photo: Mark J. Terrill, AP)

SIMI VALLEY, Calif. – President Trump?

The celebrity billionaire who showed in the first Republican debate six weeks ago that he ought to be taken seriously as a potential nominee walked on stage for the second debate Wednesday determined to demonstrate that now he should be taken seriously as a potential president.

Donald Trump's rivals were equally determined to prove that wrong, raising questions about his conservatism, his temperament and his depth of knowledge on national security and other issues.

"Everything I've done, virtually, has been a tremendous success," Trump declared as he dismissed rivals who said he wasn't suitable to serve as commander in chief. He and the other 10 participants at the debate, sponsored by CNN and held at the Reagan Presidential Library, were standing in front of a jetliner used as Air Force One during Ronald Reagan's presidency. Citing his record in making billions, writing the best-selling Art of the Deal and hosting TV's The Apprentice, he said: "What I want to do is put that ability in this country to make this country great again."

The others weren't convinced.

Kentucky Sen. Rand Paul called Trump "sophomoric" and asked, "Do we want someone with that kind of character, that kind of careless language to be negotiating with (Russian President Vladimir) Putin, to be negotiating with Iran?" Former Florida governor Jeb Bush, asked about Trump, said fixing the nation's "extraordinarily difficult" problems "requires a steadiness, that requires an understanding of how the world works. You can't just, you know, talk about this stuff and insult leaders around the world and expect a good result."

"Our leading candidate gets his foreign policy from watching television," South Carolina Sen. Lindsey Graham said in the undercard debate of four contenders whose low poll standings didn't get them in the main event. "That's not foreign policy; that's a cartoon character."

That reflects a change of strategy since the first debate, when most of his competitors were wary of taking on Trump. Then, they calculated that his bombast and penchant for outrageous statements would catch up with him — that, in effect, he would bring down himself.

That hasn't happened. Even Trump's suggestion that Fox News anchor Megyn Kelly might have asked tough questions because of hormones didn't dent his standing — or prompt an apology. In the current RealClearPolitics averages of recent polls, Trump leads retired neurosurgeon Ben Carson, his closest competitor, by 5 percentage points in Iowa, 15 points in New Hampshire and 11 points nationwide.

Take a step back to consider that state of affairs: A reality TV star and real-estate mogul mocked as The Donald has managed to build and hold a significant lead over the summer among likely Republican voters nationwide and in the states that hold the first two contests next year.

Meanwhile, the more credentialed contenders who were viewed as credible nominees from the start have had to rethink their approaches and reassure their donors as they sink in surveys. Bush, whose fundraising clout and family connections haven't been enough to keep him out of single digits, faced the problematic imperative during the debate to disprove Trump's jibe that he is "low energy." (Bush said his Secret Service code name should be "Ever Ready.")

Donald Trump, right, and Ben Carson walk onstage during the presidential debate at the Reagan library on Sept. 16, 2015, in Simi Valley, Calif. (Photo: Sandy Huffaker, Getty Images)

That said, the hurdles now get higher for Trump. (He jokingly said his Secret Service code name should be "Humble.") The debate Wednesday marked a new phase of the campaign. After a "Summer of Trump," an onslaught of TV ads financed by candidates and their super PACs will begin to air; organizational efforts will intensify and the record-sized field will begin to be sorted out.

For another, all the candidates face sharper questions about what, precisely, they would do as president. That applies to Trump, who up to now has relied more on sweeping declarations — "We have to build a wall" — than detailed policies. One of the questioners at the CNN debate, conservative radio talk-show host Hugh Hewitt, stumped Trump in an interview last week when the candidate confused the Iran's Qud forces with the Kurds, and couldn't distinguish between Hamas and Hezbollah.

In the debate, former Hewlett-Packard CEO Carly Fiorina referred to Trump's stumble in that interview when the candidates were asked about policy toward Russia, then pointedly ticked off a series of specific steps she would take to curtail Putin. Trump answered in much more general terms. "First of all, they have to respect you," he said, and went on, "I would get along with a lot of the world leaders that this country is not getting along with."

Fiorina, by the way, got some of the most sustained applause of the evening, including when she declined to accept Trump's explanation that he was referring to her "persona" when he told an interviewer, "Look at that face!" in describing Fiorina as an unelectable candidate. "I think women all over this country heard very clearly what Mr. Trump said," she said unsmilingly.

Trump has made inroads in convincing Republicans he has the qualities and experience a president needs, but he continues to face wide skepticism from others. In an ABC News-Washington Post Poll released this week, Republicans by close to 2-1 said Trump was "qualified to serve as president," but Americans overall by nearly 2-1 said he wasn't.

A majority of Republicans said Trump had "the kind of personality and temperament it takes to serve effectively as president," but Americans overall by nearly 2-1 said he didn't.

During the debate, Trump made faces, sighed loudly and made interjections — "Wrong!" — when others criticized him. He rejected Bush's demand that he apologize to Bush's Mexican-born wife for a comment about her that he had made on the campaign trail. When Paul blasted Trump for ridiculing others on their looks, he pointed at the Kentucky senator and protested, "I never attacked him on his look, and believe me, there’s plenty of subject matter there." When Bush tangled with Trump over whether, as governor, Bush had prevented Trump from opening casinos in Florida, Trump delivered a back-handed compliment.

"You're showing more energy tonight," he said patronizingly. "I like that."

Bush was briefly speechless.

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