Can I always think of derivatives this way?

[Ahmed1029]

TL;DR Summary

Can I always treat derivatives as a small element lf y divided by a small element of x?

In physics, the differential is always treated as a little change or a tiny element of something, be it volume, area, etc. However, when I differentiate a function with respect to another, I always of it as a change divided by a change, not an element divided by an element: like when volume is an explicit function of area, I think of how much the volume changes, rather than taking a small element of volume and dividing it by a small element of area. My question is : are the two "definitions" equally valid? Can I always think of a derivative both ways? + I only know calculus and elementary differential equations so I will probably not understand any answer given in terms of differential forms, so please keep it simple

 

[fresh_42]

The problem with the infinitesimal picture is that $\dfrac{dx}{dt}=\displaystyle{\lim_{t\to 0}\dfrac{x(t_0+t)-x(t_0)}{t}}$ has two quantities on the left but only one limit on the right. Pretending $dx,dt$ were two limits gets immediately wrong.

 

[fresh_42]

https://www.physicsforums.com/threa...simal-x-when-we-neglect-x-2-in-x-x-2.1017135/
https://www.physicsforums.com/threads/reasoning-behind-infinitesimal-multiplication.1016639/

are two recent discussions about it.