Thanks. I hope that someday I could be helpful too here.(Blush)
Actually, it is "Derivative" which is rigorously defined -not "Differential"- however if \frac{\mathrm{d} f(x)}{\mathrm{d} x} was just a notation for derivative, there would be no problem. But we go further and define differential...
I understand that Leibniz's notation has so many intuitive and practical advantages that help students who have not much interest in higher mathematics to solve calculus or differential equations problems easier. But it would be better if we could keep rigor and intuition at the same time...
In this case, I think it's really bad to use Leibniz's notation in calculus. (Crying) We should reformulate Calculus using Lagrange's notation for both derivative and antiderivative operations so we get rid of this nonsense differential. (Sadface)
First of all I want to clarify that I posted this question on many forums and Q&A websites so the chances of getting an answer will be increased. So don't be surprised if you saw my post somewhere else.
Now let's get started:
When it comes to definitions, I will be very strict. Most textbooks...