Question about the meaning of derivative .

[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.

 

[rollingstein]

Sounds good to me broadly.

 

[thewoodpecker]

Hi, thanks for the reply.
Please if someone see something wrong in my statement, correct it.

 

[Mark44]

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.

 

[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.

 

[Vorde]

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.