11 questions linked to/from Rigorous definition of "differential"
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### Why can't the second fundamental theorem of calculus be proved in just two lines?

The second fundamental theorem of calculus states that if $f$ is continuous on $[a,b]$ and if $F$ is an antiderivative of $f$ on the same interval, then: $$\int_a^b f(x) dx= F(b)-F(a).$$ The proof of ...
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### Differentials Definition

Please define differentials rigorously such that they give a consistency to their use in the following links. I have read Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? What is the practical ...
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### When can't $dy/dx$ be used as a ratio/fraction?

By searching this question, I found this: Can I ever go wrong if I keep thinking of derivatives as ratios? However, the answers don't have what I'm looking for! (Edit: Meaning, a counterexample. ...
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### Infinitesimal calculus

I have been reading some non-standard analysis from Keisler's book and I think it is logically consistent till now but there are criticisms against it and why isn't non-standard analysis accepted more ...
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### Learning differential calculus through infinitesimals

In class, we've studied differential calculus and integral calculus through limits. We reconstructed the concepts from scratch beginning by the definition of limits, licit operations, derivatives and ...