# A Different Definition of Derivative

• Unassuming
In summary, the conversation discusses the proof that if a function f is differentiable at p, then the limit of the sequence n[f(p+1/n)-f(p)] as n approaches infinity is equal to f'(p). It also considers an example where the existence of the limit of the sequence does not imply the existence of f'(p). The conversation also mentions an attempt to compare the limit to a given definition, but it did not lead to a solution. Finally, it is noted that the function must be differentiable at p for all sequences that converge to p in order for the proof to hold.
Unassuming

## Homework Statement

part 1. )

If f:(a,b)--> R is differentiable at p in (a,b), prove that

f ' (p) = lim (n -> oo) n [ f(p + 1/n) - f(p) ].

part 2. )

Show by example that the existence of the limit of the sequence,

{n [ f(p + 1/n) - f(p) ] } does not imply the existence of f ' (p).

## The Attempt at a Solution

One failed attempt involved comparing this limit to the definition below,

$$\lim_{n \rightarrow \infty} \frac{f(p_n)-f(p)}{p_n-p}$$.

I attempted to break the new definition (up top) into the sum of the limits and proceed down that path but it did not lead to anything.

I have also tried this new definition with a functions and a point and compared its result to f'(p) to gain some understanding. I still cannot seem to get anywhere though.

Any help would be appreciated. Thank you.

That sequence looks EXACTLY like your definition where p_n=p+1/n. What's the problem?

I see how this works for the sequence p_n= p + 1/n. Do we not have to show that the two are equivalent for all sequences that converge to p, and/or is that what we are doing?

If the function is differentiable at p then all such sequences converge to the derivative. For the second part they want to find a function that is NOT differentiable at p, but where the limit of that particular sequence does exist.

## 1. What is a different definition of derivative?

A different definition of derivative is a mathematical concept that describes the rate of change of a function at a specific point. It is typically represented as the slope of a tangent line to the function at that point.

## 2. How is this different from the traditional definition of derivative?

The traditional definition of derivative involves taking the limit of a difference quotient as the interval between two points approaches zero. However, a different definition of derivative involves taking the limit of a difference quotient as the interval between two points approaches a non-zero value, allowing for a more general definition of the derivative.

## 3. What are the advantages of using a different definition of derivative?

Using a different definition of derivative allows for a more general and flexible approach to understanding the rate of change of a function. It can also provide a better understanding of the behavior of functions that are not continuous or differentiable at certain points.

## 4. How is a different definition of derivative applied in real-world situations?

A different definition of derivative is commonly used in physics, engineering, and economics to model the behavior of complex systems. It can also be used in optimization problems to find the maximum or minimum values of a function.

## 5. Is a different definition of derivative widely accepted in the scientific community?

While the traditional definition of derivative is still widely used, the concept of a different definition of derivative has gained acceptance in the scientific community, especially in fields such as calculus, physics, and engineering. It is a valuable tool for exploring and understanding complex functions and their behavior.

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