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Possible Duplicate:
Where is the flaw in this argument of a proof that 1=2? (Derivative of repeated addition)

\begin{align*} x^2 &= \underbrace{x + x + x + \dots + x}_{x \text{ times}}, \\ \therefore \frac{\mathrm{d}}{\mathrm{d}x} (x^2) &= \frac{\mathrm{d}}{\mathrm{d}x} (\underbrace{x + x + x + \dots + x}_{x \text{ times}}) \\ &= \underbrace{1 + 1 + 1 + \dots + 1}_{x \text{ times}} \\ &= x. \end{align*}

But we know that $$ \frac{\mathrm{d}}{\mathrm{d}x} (x^2) = 2x. $$

So what is the problem?

My take is that we cannot differentiate both sides because $\underbrace{{x+x+x+\cdots+x}}_{x \text{ times}}$ is not fixed and thus $1$ is not equal to $2$.

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marked as duplicate by Nate Eldredge, Douglas S. Stones, Rahul, Hans Lundmark, Zev Chonoles Jun 29 '12 at 7:17

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    $\begingroup$ What do you mean by $x$ times? $\endgroup$ – Yai0Phah Jun 29 '12 at 3:55
  • $\begingroup$ Adding x, x times x+x+x+x....(x times x) $\endgroup$ – Bazinga Jun 29 '12 at 3:56
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    $\begingroup$ Worth reading. (maa.org/devlin/devlin_0708_08.html) $\endgroup$ – user17762 Jun 29 '12 at 4:03
  • $\begingroup$ Note that the terms will be finite for any finite value of $x$ $\endgroup$ – Pedro Tamaroff Jun 29 '12 at 4:05
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    $\begingroup$ $x^2 = \lfloor x\rfloor x + (x-\lfloor x\rfloor)x$. The first part, $\lfloor x\rfloor x$, is roughly like adding $x$ "$x$ times" while still adding an integer number of $x$s. The derivative of $\lfloor x\rfloor x$ is $\lfloor x\rfloor$ where it is defined. $\endgroup$ – Jonas Meyer Jun 29 '12 at 4:40
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Simply because "$x \text{ times}$" is also a "function" of $x$. One mistake is not considering that in the derivation.

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    $\begingroup$ Besides, it's also not clear what it means to add a number to itself a non-integer number of times. That interpretation of multiplication breaks down when you move beyond integers to fractions. $\endgroup$ – Neal Jun 29 '12 at 4:06
  • $\begingroup$ @Neal True. I'm thinking about it from the differentiation perspective. $\endgroup$ – Pedro Tamaroff Jun 29 '12 at 4:20
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    $\begingroup$ I'm sure that Neal is also thinking about it from a differentiation perspective. You can't take the derivative of a function with no definition, or that is defined only at the integers, in the context of calculus on the real line. $\endgroup$ – Jonas Meyer Jun 29 '12 at 4:27
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You say "$x\text{ times}$". The number of "times" you add it up---the number of terms in the sum---keeps changing as $x$ changes. An what if $x=1.6701$? How do you add up $x$ $1.6701$ times?

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  • $\begingroup$ The arrangement of math equations in the question is very ugly. Help to edit it, such as $\underbrace{x+x+\cdots+x}_{x\textrm{ times}}$ instead of the one in the question, and displaymath is not necessary. His $dy/dx$ can be arranged with inline math. $\endgroup$ – Yai0Phah Jun 29 '12 at 4:21
  • $\begingroup$ Concerning defining repeated addition for the reals, was your question for the OP's author's consideration? I think the problem is not with defining such addition, but with using the definition for $x^2$ that has been used in the OP. $\endgroup$ – ThisIsNotAnId Jul 27 '12 at 3:36

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