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This seems too easy, but here's the question:

$x^2$ is $x + x + ...+ x$ (with $x$ terms). Its derivative is $1 + 1 + ... + 1$ (also $x$ terms). So the derivative of $x^2$ seems to be $x$.

And another expression: we know that if $y = nx$, then $y' = n$, so that if $y = x * x$ then $y' = x$.

But we know by formula that if $y = x^2$, then $y' = 2x$

So, how to prove $y' = x$ is wrong ?

Thanks

  • 1
    Addition is a binary operation. You can use this to add a positive integer number of terms after making use of associativity. But you can't just add pi terms to them selves2012-11-15
  • 6
    The number of terms varies with $x$ so you need to apply the derivative process to "with $x$ terms" as well as to each of the individual terms.2012-11-15
  • 0
    Wny don't you use the definition of the derivative to see that your argument is wrong... In the expression $y=x \cdot x$ you have a product of two functions, so the formula you use for the derivation is not valid. You need to use the derivation of the product...2012-11-15
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    So $(1/2)^2$ is $1/2+\cdots+1/2$ (with $1/2$ terms)?2012-11-15
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    I think that the "real" problem lies on forgetting the derivative of "with $x$ terms". Doing the exact same reasoning for the finite difference (defined only for natural number as $f(n+1)-f(n)$) do not have the $0.5^{0.5}$ problem, but still is wrong for forgetting that you also vary the number of addend.2012-11-15

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