Yet the president shows no signs of asking her to resign, reflecting in part his lack of interest in the issue of education and the department responsible for it. And DeVos has no interest in departing. Advisers say she is excited by the tasks ahead. After two years of mostly undoing the work of her predecessors, she has shifted to advancing her own agenda.

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Topping her list is a proposal for a $5-billion-a-year tax credit that would reimburse taxpayers and corporations dollar for dollar for donations to scholarship programs. DeVos, 61, came to Washington after a lifetime of advocating for school vouchers and other programs that allow families to channel tax dollars away from traditional public schools. Passage of such a plan would represent a crowning achievement — though it is unlikely, given widespread Democratic opposition.

DeVos persuaded the Treasury Department to support the idea, even though the credit would complicate the tax code just two years after a bill passed to simplify it. She worked behind the scenes to negotiate details and unite most school choice proponents behind the plan. Now, she is traveling the country to promote the idea, with trips so far to three states and more planned.

At the White House, aides do not expect the measure to become law, and Trump hardly mentions it. But White House officials say DeVos gets credit for pushing the school choice agenda, which is popular with Trump’s core of conservative supporters.

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And DeVos, who is deeply religious, scores points for the president with evangelical Christians, an important part of his base that has stuck by Trump even as unseemly details of his personal life have spilled out.

“He has staffed his administration and surrounded himself with people who have deep roots and street cred in the faith community. Betsy would be at or near the top of that list,” said Ralph Reed, founder of the Faith and Freedom Coalition and a longtime evangelical leader.

DeVos does not shy from talking about her faith. At an event in January hosted by the Council for Christian Colleges & Universities, she spoke of her Christian education and said her faith helps her deal with public criticism.

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“There’s an audience I play to, and it’s just an audience of one,” she said. “That’s a true north star.”

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This account of DeVos’s endurance in the Education Department’s top job is based on interviews with eight people with direct knowledge of the secretary’s relationship with the president and with an understanding of the inner workings of the White House and education agency. Many of those people spoke on the condition of anonymity because of the sensitive nature of the relationships involved.

So far, officials occupying 15 Cabinet-level positions have been fired or resigned since Trump took office. Leaders at State, Justice, Defense, Homeland Security, Interior and Health and Human Services have been fired, resigned under pressure or quit in protest. Trump has also overseen enormous turnover among top White House staffers.

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DeVos has benefited from Trump’s lack of interest in education, officials say. And the president — despite his “Apprentice” reputation for dispatching with poor-performing employees — is actually loath to fire subordinates. In many cases, he’s let them dangle for months before cutting the rope or makes their lives so miserable they quit.

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DeVos aides say she has a good relationship with Trump. One adviser said the president calls DeVos maybe once a month to talk. He dismissed reports of the president speaking negatively about DeVos, saying he makes derisive remarks about all sorts of people.

Also bolstering DeVos’s standing: She hasn’t had a single personal scandal. She’s a billionaire and travels by private plane, but she pays for it herself. She donates her salary to charity. Even detractors say that in person, DeVos is pleasant and easy to be around. And she has shown personal grit, appearing in public in a wheelchair after she broke her pelvis in a cycling accident.

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In contrast, White House officials describe Trump as more hot and cold regarding DeVos and said he rarely sees her. He has been frustrated with her public mistakes, beginning with her disastrous confirmation hearing, they said, and expects perfection from his lieutenants.

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But Trump appreciates that she’s tough, handles criticism and is a loyal soldier willing to defend even unpopular policies, officials said. For instance, she spent three days last month defending the administration’s plan to eliminate nearly $18 million in federal funding for a Special Olympics program in schools. She had fought to maintain the spending and was overruled by the White House budget office but still argued for the cut before hostile lawmakers at two congressional hearings.

Then, after the three-day mini-drama, Trump swooped in and announced he was overruling “my people” and favored the funding. It prompted a rare, albeit gentle, DeVos pushback.

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“I am pleased and grateful the president and I see eye-to-eye on this issue, and that he has decided to fund our Special Olympics grant,” she said. “This is funding I have fought for behind the scenes over the last several years.”

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Before that, she had kept quiet about the internal dispute. Early in the administration, she attended a dinner for the Special Olympics, dining with athletes and then speaking about her support for the program. Two weeks later, Timothy Shriver, chairman of the Special Olympics, was shocked to see the president’s first budget plan, which proposed cutting all federal support for the group.

He called DeVos to ask about it. At first, she defended the cut but then backed down, implying it was never her idea. Six months later, she donated a quarter of her salary to the nonprofit. Congress ignored the president’s request and increased the funding.

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DeVos kept quiet on other disagreements with the White House, too. She was against revoking documents meant to help schools work with transgender students but never publicly protested. She didn’t think that a school safety commission, formed after the mass shooting in Parkland, Fla., should consider the question of racial disparities in student discipline. Again, she said nothing.

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Aides describe her as a loyal soldier, an approach that has helped keep her position with Trump secure. But DeVos has done little to win over critics who opposed her from the start. Detractors say she lacks basic knowledge about education, caring only about her pet issue of school choice. They charge that she wants to destroy, not bolster, public education. And they argue that someone who has never attended a public school has no business being education secretary.

“She is undeterred in her mission despite the forces against her,” department spokeswoman Elizabeth Hill said. “People see she is in it for the right reasons.”

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Aides said DeVos has met with Democrats who might support her tax credit plan but declined to name them. She has never reached out to Charlie Barone, lobbyist for Democrats for Education Reform, a group that favors some of the same policies — such as more charter schools — and who might have been at least an occasional ally.

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Barone noted that Democrats have supported similar tax credits at the state level but predicted most in Congress would reject a federal plan because it is not coupled with support for public schools.

“They’re crazy if they think they have a chance with Democrats on this,” he said.

Outside Washington, DeVos still confronts protesters at public events. Inside the events, she rarely engages in discussion of topics that are not part of the program.

Last fall, she toured Holmes County Central High School in Lexington, Miss., in the Mississippi Delta, a poverty-stricken region. She was there to observe a distance-learning advanced-placement physics class, aided in part by Ivy League students tutoring over Skype. The secretary gamely participated in an experiment involving flicking a piece of paper from under a penny balancing on her finger.

“Oh, I did it!” she exclaimed, high-fiving a student.

During a roundtable discussion, one person raised an issue affecting the school system — “very severe” teacher shortages. Dozens of classes were being taught by uncertified long-term substitutes because the system cannot recruit enough teachers. DeVos did not respond to that point.

When asked about the teacher shortage by local reporters, the secretary replied that teachers need more autonomy and more opportunities for advancement, and she said schools should think “outside the box.” She did not offer ideas for attracting more teachers to the rural district.

“If you’re focused on doing the right thing for students,” she said, “solutions are going to follow.”

Asked later about her response, Nathan Bailey, her chief of staff, said it’s not her role to offer specifics.