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I'v learnt that

. . .

If a single-variable and differentiable function $y=f(x) (f:R→R)$ is given,

and if a two-varible function $dy$ is defined as $$dy = (f'(x))(dx),$$

and if an another two-variable function $△y$ is defined as $$△y = f(x+dx) - f(x),$$

(Independent variables are $x$ and $dx$ in both cases)

(Conclusion) if $dx→0$, then $$(dy) - (△y) → 0.$$

. . .

It was on Stewart calculus book(chapter 'Linear approximation and differential') and other many physics books use this property as an useful property.

My question:

▶ In the same way, is it possible to approximate $(f''(x))(dx)(dx)$ to $f(x+2dx) - 2f(x+dx) + f(x)$ ?

▶ Also, can you give me some advices about where can I find the rigorous treatment to the above contents??

Although my stuwart calculus book teaches about f'(x)(dx) -> f(x+dx) - f(x), but it doesn't provide rigorous proof. It is content with merely making readers intuitively understand this property using geometrical representation with tangential line of the function. Can I find more rigorous treatments in another calculus or analysis books?

Thank you for reading my question.

CreamOfCrop
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  • Infinitesimal analysis robinson – tryst with freedom Feb 24 '22 at 21:57
  • Thank you for replying Buraian. Did you mean Nonstandard analysis by Abraham Robinson? It looks like a way more serious book than I thouhgt.. Or do you mean books like this? https://www.amazon.com/Infinitesimal-Analysis-Mathematics-Its-Applications/dp/1402007388

    Do you think should I have to look some nonstandard analysis for comprehension of this differential and approximation things?

     – CreamOfCrop Feb 24 '22 at 22:13
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    Definitely not. Read about Taylor polynomials. – Ted Shifrin Feb 24 '22 at 22:19
  • Yes Robinson, they give an insight into what 'dx' means. There is also a related notion of a differential form, but the connection between that is not clear (a question is there asking about it). In usual mathematical analysis, we don't bother with such differential stuff. Taylor polynomials are different, it's an approximation to the function by polynomials – tryst with freedom Feb 24 '22 at 22:40
  • Thank you Ted Shifrin. Thanks to Taylor series, I'v been able to manage $(dx)^2$ like an zero when $dx$ is so small that $f(x+dx)-f(x)$ becomes similar to $f'(x)(dx)$ in physics problems. But that's all I know about the relationship between Taylor series and this property. Does Taylor expansion also can be used as a proof for f'(x)(dx) -> f(x+dx)-f(x)? Like as a premise for this conclusion? – CreamOfCrop Feb 24 '22 at 22:40
  • https://buraian.medium.com/a-brief-introduction-to-taylor-series-47416e7b9123 , here is an article on it which I have written – tryst with freedom Feb 24 '22 at 22:40
  • My advice would be to just run with it. The differential is just a quick and dirty way to talk about how asymptotic behavior. The idea of differential being the finite difference in a limit will serve you well for pretty much most purposes you encounter in classical physics and more. But, just keep in the back of your mind that the actual description of what it is requires more stuff like the hyper real numebrs – tryst with freedom Feb 24 '22 at 22:43
  • https://www.youtube.com/watch?v=ceaNqdHdqtg it maybe nice to look at this different number system as well – tryst with freedom Feb 24 '22 at 22:45
  • One physics field where differentials take main stage is Thermodynamics. There is an MIT OCW by Moungi bawendi, the mathematics done it is based on differentials. So, if you want 'physically motivated' treatment of differentials without getting into technical details (like how they work, what you can do with them etc), then that's a good start. One last thing: The number system ideas is even more general, there turns out to be other number system arising from modelling simple games : https://www.youtube.com/watch?v=ZYj4NkeGPdM. – tryst with freedom Feb 24 '22 at 22:48
  • Sorry for my late replay Buraian. I'v been checking whether (f''(x))(dx)(dx) is able to be approximated to f(x+2dx)−2f(x+dx)+f(x) or not when dx is small as in my question using python. Consequently, it looks right approximation even though I can't prove it rigorously. I have to gonna look link you gave me. Also, I decided to read a textbook using robinson's hyperreal method.(Though it's not robinson's book cause I think I need an easier one: Infinitesimal Calculus - James Henle) – CreamOfCrop Feb 25 '22 at 11:45
  • @CreamOfCrop: The other commenter was wrong; non-standard analysis is not worth learning in almost all cases, and the set-theoretic assumptions needed to set it up have no relevance to the real world. Take a look at the posts linked from my profile on "general differentiation" and "asymptotic expansion", for a proper answer to your questions. – user21820 Jun 25 '22 at 20:17

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