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How exact is your Tell exactly how small quantity must dx be so that (dx)^2 can be ignored.

What if we make a change equal to dx to the variable x? Doesn't the function f(x) change equal to f(x + dx) ?

Who told you about \(\frac{dy}{dx}\) but hided the information about \(\frac{f(x+dx)- f(x)}{dx}\) ? Was it me or was it all the math books? And why should they, the mathematicians, have done so? Why can't "usual mathematics" deal with something that is infinitely small? What is the question? Is the question "is 1/∞ either 0 or non-zero?" ? The question does not have an unambiguous answer either 0 or non-zero, because 1/∞ is both. If that is a stupid thing to do, then why don't you answer the question: "is 1/∞ either 0 or non-zero?" ? Yes. But I am not talking about sets, because maybe it is not necessary right now. I am talking about finite difference f(x+h) - f(x) where h is finite, as opposed to f(x + dx) - f(x) where dx is not finite. Adjective finite refers to distance in this case, finite distance. You can count the elements of X only if you know n. But because you did not tell what is n, I can't tell how many elements are there in X. So it may look as though dx is a set for which you don't tell what is n.

It makes no difference because y = f(x) so that dy/dx = df(x)/dx 1/∞ is infinitely small. Is it 0 or non-zero? The introduction of infinity brings a duality into the definition of an infinitesimal, meaning that we have to deal with objects that both zero and non-zero at the same time.

Yes. Derivative is defined as the limit of a finite difference \[\frac{dy}{dx} = \lim_{h\to\ 0}\frac{f(x+h)- f(x)}{h}\] where h is finite. I mean that dx is not finite as opposed to h which is finite. If dx is not finite, it is infinite, infinitely small.

Derivative is defined as the limit of a finite difference \[\frac{dy}{dx} = \lim_{h\to\0}\frac{f(x+h)- f(x)}{h}\] where h is finite. Is it possible to define the derivative by \[\frac{dy}{dx} = \frac{f(x+dx)- f(x)}{dx}\] where dx is not finite but infinitely small, infinitesimal ?

\[\frac{\Delta y}{\Delta x}= \frac{f(x+h)- f(x)}{h} \] \[\frac{\Delta y}{\Delta x} = \frac{f(x+h)- f(x)}{h}\] \[\Delta y \] \[ y = \int f(x) dx \] \[\Delta y \] \[ y = \int f(x) dx \] I got it working now, except the last one , integral doesn't like me. Now it all works. I noticed there was a message saying something like "pasted as rich text, paste as plain text instead" which I choose and then the integral also worked.