Question about the meaning of derivative .

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In summary, the derivative at a given point is the limit of the difference quotient as Δx goes to zero. This limit does not change as Δx gets close to zero.
  • #1
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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  • #2
Sounds good to me broadly.
 
  • #3
Hi, thanks for the reply.
Please if someone see something wrong in my statement, correct it.
 
  • #4
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.
 
  • #5
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.
 
  • #6
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.
 

1. What is a derivative?

A derivative is a mathematical concept used to describe the rate of change of a function at a specific point. It represents the slope of the tangent line to the function at that point.

2. How is a derivative calculated?

A derivative is calculated using the limit definition, which involves finding the slope of a secant line as the two points get closer and closer together. This limit is represented by the notation f'(x) or dy/dx.

3. What is the significance of derivatives?

Derivatives have many real-world applications, such as in physics, economics, and engineering. They are used to analyze rates of change, optimize functions, and solve various problems.

4. What is the relationship between derivatives and integrals?

Derivatives and integrals are inverse operations of each other. The derivative of a function is its slope, while the integral of a function is the area under its curve. The Fundamental Theorem of Calculus states that integration and differentiation are opposite processes.

5. How do derivatives relate to the concept of instantaneous rate of change?

A derivative represents the instantaneous rate of change of a function at a specific point. This means it shows how much the function is changing at that exact moment, rather than over an interval. This concept is useful in understanding the behavior of a function at a particular point.

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