Question about the meaning of derivative .

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Discussion Overview

The discussion revolves around the concept of the derivative in calculus, specifically focusing on its definition and interpretation. Participants explore the meaning of the derivative as the limit of the difference quotient and the implications of considering points that are infinitely close together.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant states that the derivative at a given point is the limit of the difference quotient as Δx approaches zero.
  • Another participant agrees broadly with this definition but seeks confirmation on their understanding.
  • A participant questions the phrase "distance as one point," suggesting that it might imply a misunderstanding of how distance is defined.
  • There is a proposal that two points can be viewed as one point if they are infinitely close, which is met with a challenge regarding the nature of infinitesimals.
  • A later reply emphasizes that while infinitesimals are close to zero, they are not zero, which is crucial for the mathematical operations involved in defining a derivative.

Areas of Agreement / Disagreement

Participants generally agree on the basic definition of the derivative but express differing interpretations regarding the implications of infinitesimals and the concept of points being infinitely close together. The discussion remains unresolved regarding the interpretation of these concepts.

Contextual Notes

There are limitations in the discussion regarding the precise definitions of terms like "infinitely small" and "distance," as well as the mathematical treatment of infinitesimals. These aspects are not fully clarified or agreed upon.

thewoodpecker
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Hello

I know that the definition of a derivative at given point is the limit of the difference quotient as Δx goes to zero.

I just want to be sure, that I have understood it right. So i have this question.Is the derivative at a given point is the ratio of change of dependent variable and change of independent variable over a so small distance (infinitely small) that we can assume this ratio(slope) does not change in that distance, and we can look at this distance as one point.
 
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Sounds good to me broadly.
 
Hi, thanks for the reply.
Please if someone see something wrong in my statement, correct it.
 
thewoodpecker said:
Hello

I know that the definition of a derivative at given point is the limit of the difference quotient as Δx goes to zero.

I just want to be sure, that I have understood it right. So i have this question.


Is the derivative at a given point is the ratio of change of dependent variable and change of independent variable over a so small distance (infinitely small) that we can assume this ratio(slope) does not change in that distance, and we can look at this distance as one point.
What does "distance as one point" mean? Distance is a measure of how far one point is from another point.
 
Hi, thanks for the reply.
Would it be correct if i say the two points are infinitely close to each other so we can view at these two points as one point? The distance between them is infinitely small.
Thanks for help.
 
thewoodpecker said:
Hi, thanks for the reply.
Would it be correct if i say the two points are infinitely close to each other so we can view at these two points as one point? The distance between them is infinitely small.
Thanks for help.

No. The key here was in your original post: "goes to zero." The key thing about infinitesimals is that they may be infinitely close to zero, but they are not zero. If they were, then you couldn't divide by them.

The way you should look at it is that you are considering the limit as the difference between two points gets infinitely close to zero but does not become zero.
 

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