Patrick, interviewed at the Kentucky Speedway, told ESPN there are "going to be people who believe in you and people who don't."
"Plenty of people say bad things about me. I see it on Twitter," she said. "Some people want me to die. But at the end of the day, you get over that stuff and trust you're doing a good job for the people who believe in you."
Petty, son of seven-time NASCAR champion Richard Petty and a former NASCAR driver himself, called Patrick a "marketing machine" and said she is "not a race car driver." Petty, now an analyst for Fox News and TNT, spoke on Speed TV's "NASCAR Race Hub."
"Danica has been the perfect example of somebody who can qualify better than what she runs," Petty said. "She can go fast, but she can't race. I think she's come a long way, but she's still not a race car driver. And I don't think she's ever going to be a race car driver."
Patrick said she found his comments amusing.
"It's funny he said I can qualify and not race," she said. "Anyone who has watched what I do knows I'm crap in qualifying. In the races, things go much better."
Patrick, 31, has been having a difficult time as a Sprint Cup driver, ranking 27th. But she has six Top-10 finishes in the Indy 500, coming in third in 2009, the top finish ever by a woman.
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