Find the fallacy in the derivative.

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In summary, the conversation discusses a problem understanding conflicting results of a derivative and identifies a fallacy in deriving x^2 as a sum of x's. The summation is not well defined as a function from R to R and the number of terms cannot change in the summation rule. The function f(x) is not well defined between R and R if there exists a real value x for which f(x) is not defined.
  • #1
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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  • #2
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.
 
  • #3
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?
 
  • #4
Writing x^2 as a "sum of x 'x's" requires that x be a positive integer, not a general real number.
 
  • #5
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.
 

1. What is a fallacy in the derivative?

A fallacy in the derivative refers to an error or mistake in the calculation of the derivative of a function. It is a common error in mathematics and can lead to incorrect results.

2. How can I identify a fallacy in the derivative?

A fallacy in the derivative can be identified by carefully checking the steps used to calculate the derivative. Look for any mistakes in algebraic manipulation, application of derivative rules, or incorrect use of notation.

3. What are the consequences of a fallacy in the derivative?

A fallacy in the derivative can lead to incorrect values for the derivative of a function, which can then affect any subsequent calculations or applications of the derivative. It is important to identify and correct any fallacies to ensure accurate results.

4. How can I avoid making a fallacy in the derivative?

To avoid making a fallacy in the derivative, it is important to carefully follow the rules and steps for calculating derivatives. Double-checking your work and using multiple methods to verify the result can also help catch any potential errors.

5. Are there common types of fallacies in the derivative?

Yes, there are several common types of fallacies in the derivative, such as applying the power rule incorrectly, not using the chain rule when necessary, and making mistakes in algebraic simplification. Being aware of these common mistakes can help in identifying and avoiding them.

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