To get the function dy/dx for say y = xx, you do
y + del y = (x+del x)(x + del x), where del y is the increment to y when x is incremented by del x.
Expanding RHS and subtracting y = xx from above we get
del y = 2x del x + del x*del x
As del x tends to zero, reaching the limit when it is very nearly zero
del y = 2x del x + del x*del x, where in comparison the latter term in the RHS is an extremely small number
As del y and del x are not zero, but only very nearly so, being infinitesimals we can have the term
del y/del x as something which is NOT incalculable like 0/0 but which is
del y/del x = 2x, which is a division process of the infinitesimals.
But as they are,and have to be, infitesimals, not just increments, which could be arbitrary, the term del y/del x
is written as
dy/dx meaning the differentiation, or slope, of the function y=f(x).

> Only the combined symbol dy/dx considered as a whole
> was allowed to have meaning.

Yes, and that is the slope of the function f(x) at the point x,y.

Hope this make things clearer. Yes, philosophy comes in hugely here, what after all is 0/0 or infinity/infinity? The whole universe, of course.
Cheers,
Arindam Banerjee, professional engineering mathematician for 45 years and going...

On Sunday, April 16, 2023 at 5:07:21 PM UTC-7, banerjee...@gmail.com wrote:
> To get the function dy/dx for say y = xx, you do
> y + del y = (x+del x)(x + del x), where del y is the increment to y when x is incremented by del x.
> Expanding RHS and subtracting y = xx from above we get
> del y = 2x del x + del x*del x
> As del x tends to zero, reaching the limit when it is very nearly zero
> del y = 2x del x + del x*del x, where in comparison the latter term in the RHS is an extremely small number
> As del y and del x are not zero, but only very nearly so, being infinitesimals we can have the term
> del y/del x as something which is NOT incalculable like 0/0 but which is
> del y/del x = 2x, which is a division process of the infinitesimals.
> But as they are,and have to be, infitesimals, not just increments, which could be arbitrary, the term del y/del x
> is written as
> dy/dx meaning the differentiation, or slope, of the function y=f(x).
>
>
> > Only the combined symbol dy/dx considered as a whole
> > was allowed to have meaning.
>
> Yes, and that is the slope of the function f(x) at the point x,y.
>
> Hope this make things clearer. Yes, philosophy comes in hugely here, what after all is 0/0 or infinity/infinity? The whole universe, of course.
> Cheers,
> Arindam Banerjee, professional engineering mathematician for 45 years and going...

Zero math is my win...