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.