Find the fallacy in the derivative.

[Flexington]

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?

 

[ebola1717]

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.

 

[Flexington]

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?

 

[HallsofIvy]

Writing x^2 as a "sum of x 'x's" requires that x be a positive integer, not a general real number.

 

[gb7nash]

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.