Can I always think of derivatives this way?

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SUMMARY

The discussion centers on the interpretation of derivatives in calculus, specifically contrasting the infinitesimal approach with the limit definition. Participants emphasize that while both perspectives can be valid, the limit definition, represented by the formula ##\dfrac{dx}{dt}=\displaystyle{\lim_{t\to 0}\dfrac{x(t_0+t)-x(t_0)}{t}}##, is more rigorous. The conversation highlights the potential pitfalls of treating infinitesimals as separate limits, which can lead to misunderstandings in calculus. Overall, the consensus is that understanding derivatives as changes divided by changes is a more reliable method for those familiar with basic calculus.

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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 🥲
 
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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.
 
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