This is a tricky one, since there's a combination of two things:
The student is probably genuinely interested in the math and wants to grapple with the material.
The student is disrupting the classroom, and interrupting the lesson for other students.
On the one hand, you want to foster the student's interest and enjoyment of mathematics. On the other, you can't allow a single student to disrupt the lecture repeatedly. I think as others have suggested, a somewhat delicate approach is needed.
I recommend that the next time the student does something along these lines, don't react angrily or allow them to fluster you, but instead in a very calm way slowly walk through the derivation and show why the student is incorrect. If it can be done with elementary methods, use them. But if not, don't necessarily shy away. The goal is simply to get to the truth. The student needs to recognize two things: (1) you, the instructor, are an expert on the material, (2) they, the student, still have a lot to learn. Being slow and methodical will hopefully make the student realize just how disruptive they're being, and responding calmly will help defuse any desire they have to elicit a reaction from you.
Having been an instructor for many freshman non-major math courses, in my experience many of the students are still very much like high-schoolers, unfortunately. But they're at an age where they want to be seen as more mature. If you keep the level of classroom maturity high, they'll respond well. But if you get dragged into a conflict with a student, it will only get worse.
Lastly, I think it's important to remember that this is only one student among many. If they're innocently asking many questions out of curiosity, it's definitely not malicious and shouldn't be treated as such. But if you get the impression that they're trying to disrupt the class for their own enjoyment, that's a serious problem, since they're detracting from the learning experience of the entire class, and I would have zero patience for students like that. If you get the impression it's the latter, I wouldn't feel badly about reporting the student to the professor. Everyone else (or, more realistically, their parents) is paying good money for the opportunity to learn in that class.