Leibniz notation makes no sense?

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    Leibniz Notation
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Discussion Overview

The discussion revolves around the understanding and interpretation of Leibniz notation in calculus, particularly in the context of derivatives and their applications such as implicit differentiation and u-substitution. Participants express confusion regarding the meaning of treating derivatives as fractions and the implications of this treatment.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses frustration with Leibniz notation, questioning its clarity and utility in understanding derivatives.
  • Another participant affirms that treating derivatives as fractions is valid and indicates that the change in one variable can be expressed in terms of another.
  • A different participant argues that while the derivative is not a fraction, it is useful to treat it as one due to its limit-based definition.
  • Another contribution mentions that there is an interpretation in differential geometry that provides a more abstract understanding of the fraction, which can be comprehensible to humans.
  • A historical perspective is introduced, referencing Newton's caution against viewing derivatives as proper fractions, using an analogy involving limits and the behavior of fractions as variables approach infinity or zero.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the clarity and utility of Leibniz notation. There are competing views on whether it should be treated as a fraction and the implications of doing so, indicating an unresolved discussion.

Contextual Notes

Participants express varying degrees of understanding and interpretation of Leibniz notation, with some acknowledging its usefulness while others remain confused. The discussion includes references to limits and abstract interpretations that may not be universally accepted.

Pithikos
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I know this question has been out there many times. I read many threads already but I just didn't find a satisfactory answer. What some people say is that Leibniz notation is just a notation and not a fraction. Then we treat this notation as a fraction. But what's the reason to do it if a human can't understand it?

I study at the moment for my test and there is implicit differentiation, u-substitution etc and I can't learn them because I am stuck with this filthy notation.

say I have s(t)=10t2

its derivative is \frac{ds}{dt}=20t

I can treat that as a fraction and get ds=20t*dt

But what the heck does that mean?? The change of s is 20t times the change of t?
 
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Hi Pithikos! :wink:
Pithikos said:
I can treat that as a fraction and get ds=20t*dt

But what the heck does that mean?? The change of s is 20t times the change of t?

Yes! :smile:

What is worrying you about that? :confused:​
 
Who told you a human can't understand it?

The first derivative of a function is NOT a fraction but it is a useful notation to write it like one because it can be treated like a fraction.

Specifically, the derivative is the limit of a fraction. To prove that the derivative has some specific property of a fraction, go back before the limit, use the property for the difference quotient, then take the limit again.
 
These answers are correct, but if it makes you feel any better, there is also an interpretation, to be given later in differential geometry, by which the fraction also makes sense as an actual fraction of things a little more abstract than numbers, and which can be understood by humans.
 
In his work on fluxions (i.e, derivatives), Newton warned about regarding them as proper fractions, and used the following analogy.

Suppose we look at the following proper fraction 2x/x=2, for some number x.
If we let x "grow" into infinity, the quantity will still equal 2, but we cannot any longer say that it is written in terms of a fraction between two NUMBERS (infinity not being a number)
Similarly the other way, by letting x shrink to 0, we no longer have any fraction to speak of, only the limit of such fractions.
 

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