Find the fallacy in the derivative.

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Discussion Overview

The discussion revolves around the confusion regarding the derivative of the function \( f(x) = x^2 \) when expressed as a sum of \( x \)'s. Participants explore the implications of defining \( x^2 \) in terms of a summation and the resulting discrepancies in derivative calculations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant presents a derivation of the derivative of \( x^2 \) as \( 2x \) and contrasts it with a summation representation leading to a derivative of \( x \), questioning the fallacy in this reasoning.
  • Another participant points out that the function \( f \) defined as a sum of \( x \)'s is only valid for natural numbers, thus not well-defined for real numbers.
  • A participant expresses confusion about the implications of a function being "not well defined" and seeks clarification on this concept.
  • Further clarification is provided that writing \( x^2 \) as a sum of \( x \)'s necessitates \( x \) being a positive integer, which limits its applicability.

Areas of Agreement / Disagreement

Participants generally agree that the summation approach to defining \( x^2 \) is problematic due to the requirement of \( x \) being a positive integer. However, there remains some uncertainty regarding the implications of this limitation and the concept of a function being well-defined.

Contextual Notes

The discussion highlights limitations in the definitions used, particularly regarding the domain of \( x \) and the application of differentiation rules. There are unresolved questions about the nature of the function and its validity across different number sets.

Who May Find This Useful

This discussion may be useful for students and individuals interested in calculus, particularly those grappling with the concepts of derivatives, function definitions, and the implications of different mathematical representations.

Flexington
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Ok,
So i have a problem understanding conflicting results of a derivative,
Consider the derivative of x2, which is 2x.
However, if x2 is expressed as a sum of x's such that f(x) = x + x + x + x ... (x times), the derivative of f(x) becomes = 1 + 1 + 1 + 1 ... (x times.) = x Hence the derivation shows the derivative of x2 to be x.
Clearly this can't be correct. Where is the fallacy in this?

My idea is that the summation is linear in X whilst x2 is non linear hence the summation won't converge to x2. However this is only an idea?
 
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Well f is only defined if x is a natural number, so it's not well defined as a function from R to R. Also, we can't apply the summation rule if the number of terms is changing.
 
Thank you for the reply.

So the derivative of the sum of x's is invalid as x is undefined.i.e. the number of terms. Its time i brushed up on the rules of differentiation. Also you say the function is not well defined between R and R, can you explain what this means as i don't understand?
 
Writing x^2 as a "sum of x 'x's" requires that x be a positive integer, not a general real number.
 
Flexington said:
Thank you for the reply.

So the derivative of the sum of x's is invalid as x is undefined.i.e. the number of terms. Its time i brushed up on the rules of differentiation. Also you say the function is not well defined between R and R, can you explain what this means as i don't understand?

If a function f(x) is not well defined between R and R, there exists a real value x such that f(x) is not defined. For instance:

f(x) = 1/x is not defined for x = 0.
 

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