Question about the meaning of derivative .

by thewoodpecker
Tags: derivative, meaning
 P: 3 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.
 P: 301 Sounds good to me broadly.
 P: 3 Hi, thanks for the reply. Please if someone see something wrong in my statement, correct it.
Mentor
P: 19,704

Question about the meaning of derivative .

 Quote by thewoodpecker 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.
 P: 3 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.
P: 781
 Quote by thewoodpecker 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.

 Related Discussions Special & General Relativity 7 Calculus & Beyond Homework 1 Calculus & Beyond Homework 2 Calculus 4 Classical Physics 1
The text passage is a discussion thread about the concept of a derivative in calculus. The original poster asks for clarification on their understanding of the derivative, and other users provide feedback and further explanation.
no_question

while the passage contains a discussion about the concept of a derivative, it does not contain a specific mathematical question that requires a numerical or symbolic answer. the questions posed are more about understanding and clarification rather than solving a particular mathematical problem.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Question about the meaning of derivative .

by thewoodpecker
Tags: derivative, meaning
 P: 3 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.
 P: 301 Sounds good to me broadly.
 P: 3 Hi, thanks for the reply. Please if someone see something wrong in my statement, correct it.
Mentor
P: 19,704

Question about the meaning of derivative .

 Quote by thewoodpecker 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.
 P: 3 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.
P: 781
 Quote by thewoodpecker 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.

 Related Discussions Special & General Relativity 7 Calculus & Beyond Homework 1 Calculus & Beyond Homework 2 Calculus 4 Classical Physics 1
user
The text passage is a discussion thread about the concept of a derivative in calculus. The original poster asks for clarification on their understanding of the derivative, and other users provide feedback and further explanation.

TYPE: NO_QUESTION

While the passage contains a discussion about the concept of a derivative, it does not contain a specific mathematical question that requires a numerical or symbolic answer. The questions posed are more about understanding and clarification rather than solving a particular mathematical problem.