What is the Meaning of dy/dx in Nonstandard Calculus?

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Discussion Overview

The discussion revolves around the interpretation of the notation dy/dx in the context of nonstandard calculus. Participants explore its meaning, particularly whether it represents a division of dy by dx or serves as a notation for derivatives in a broader sense.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that dy/dx is the Leibniz notation for the derivative, suggesting it represents the derivative with respect to y.
  • Others argue that dy/dx can be interpreted as dy divided by dx, particularly in the framework of nonstandard calculus where infinitesimals are used.
  • A participant mentions that dy/dx means the same as f'(x), indicating that it is the derivative of y with respect to x, and discusses implicit differentiation in this context.

Areas of Agreement / Disagreement

Participants express differing views on whether dy/dx should be understood as a division of dy by dx or merely as a notation for derivatives. The discussion remains unresolved with multiple competing interpretations present.

Contextual Notes

Some claims depend on the definitions of calculus being used, particularly the distinction between standard and nonstandard approaches. The implications of using infinitesimals versus limits are not fully explored.

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micromass said:
It's the Leibniz notation for the derivative. See http://en.wikipedia.org/wiki/Derivative
very long article.
Actually the thing i want to ask is Does dy/dx means dy divided by dx.
 
vkash said:
very long article.
Actually the thing i want to ask is Does dy/dx means dy divided by dx.


Not at all! dy/dx is simply a notation. It's a notation for the derivative with respect to y.
 
It means the same thing as f'(x) does. Take whatever y is and take the derivative of it with respect to x.
If we have the equation, [tex]{y=x^2}[/tex] what's the derivative with respect to x? We have to use implicit differentiation because you can't the derivative of y when we are taking it it with respect to x. So you have [tex]{1*dy/dx=2x}[/tex]

Hopefully my explanation is clear haha.
 
It does, in some sense, mean [itex]\mathrm dy[/itex] divided by [itex]\mathrm dx[/itex] in nonstandard calculus, where we use infinitesimals instead of limits to do calculus.
 

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