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**TL;DR Summary:**I attempt to find the derivative of uv with respect to x using non standard analysis, hyperreals, and the standard part function st; I take u to be a function of x, and I also take v to be a function of x.

Hello everyone!

I've been learning about non standard analysis concepts (shout-out to @Dale for clarifying a lot of concepts and taking time out of his day to help me) and thought I would try something super basic with what I learned to see if I'm actually understanding the concepts correctly, I worked out a problem, and my AI assistant objected to what I thought was a correctly worked out solution, so I wanted to double check here to see if my logic is correct, please feel free to critique my approach and show any misconceptions I may have.

My solution:

d(uv)/dx = st(((u+du)(v+dv) - uv)/dx) = st(((uv) + (u)(dv)+(v)(du)+(du)(dv)-(uv))/dx) = st((u)(dv/dx)+(v)(du/dx) + ((du)(dv)/dx)) = (u)(dv/dx)+(v)(du/dx)

Some underlying assumptions:

I assume that given some infinitesimal e and certain non zero real valued variables a, b, c, that du = ae, dv = be, and dx = ce.

I assume we can distribute st over a summation, i.e, I assume st(p1 + p2 + p3) = st(p1) + st(p2) + st(p3) where p1, p2, and p3 are parameters.

I assume that we can round the hyperreal in the analysis by assuming st((du)(dv)/dx) is zero.

Because I'm using hyperreals I'm pretty sure that means the infinitesimal e is not nilpotent.

If you feel you can shed some light, please feel free to reply, thank you!

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