Definition of the Derivative using delta and epsilon

In summary, the conversation discusses the formulation of the derivative and how it can be written using the \epsilon-\delta method. The person asking the question is seeking clarification on whether the term "limit" and the notation "lim" have the same meaning in this context.
  • #1
jfy4
649
3
Hi,

I have a question about the formulation of the derivative. The definition is

[tex]f'(x_0)=\lim_{x\rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}[/itex]

Lets say this limit exists. Can I write the limit in the typical [itex]\epsilon-\delta[/itex] method as such

Given the limit exists, then for all [itex]\epsilon>0[/itex] there exists a [itex]\delta>0[/itex] such that [itex]|x-x_0|<\delta\implies[/itex]
[tex]\left|\frac{f(x)-f(x_0)}{x-x_0}-f'(x_0)\right|<\epsilon[/tex]

?
 
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  • #2
Yes, the word "limit" and the "lim" notation mean just the same here as they usually do.
 
  • #3
Rasalhague said:
Yes, the word "limit" and the "lim" notation mean just the same here as they usually do.

Thanks
 

FAQ: Definition of the Derivative using delta and epsilon

What is the definition of the derivative using delta and epsilon?

The derivative of a function at a given point is defined as the limit of the difference quotient as the input approaches that point. This limit can be expressed using the symbols delta and epsilon, where delta represents a small change in the input and epsilon represents a small change in the output.

How is the derivative using delta and epsilon related to the slope of a tangent line?

The derivative using delta and epsilon can be interpreted as the slope of the tangent line to the graph of the function at the given point. This is because as the delta (change in input) approaches 0, the difference quotient approaches the slope of the tangent line.

What is the significance of using delta and epsilon in the definition of the derivative?

Using delta and epsilon in the definition of the derivative allows us to precisely define the concept of instantaneous rate of change, which is the fundamental concept of calculus. It also allows us to generalize the concept of the derivative to functions that are not continuous or differentiable at every point.

How is the definition of the derivative using delta and epsilon related to the limit definition of the derivative?

The definition of the derivative using delta and epsilon is essentially a more formal and rigorous way of expressing the limit definition of the derivative. It uses the symbols delta and epsilon to represent the small changes in input and output, rather than using the notation of h approaching 0.

What are the limitations of using the definition of the derivative using delta and epsilon?

The main limitation of using this definition is that it can be quite complex and difficult to work with, especially for more complicated functions. It also requires a thorough understanding of limits and epsilon-delta proofs. For practical purposes, it is often more convenient to use other methods, such as the power rule or the chain rule, to find derivatives.

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