# Can I always think of derivatives this way?

• I
• Ahmed1029

#### 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 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.

• jedishrfu

• jedishrfu