lin00b: My suspicion is that the mismatch occurs because you sum the x's x times, and that x times summing up of x's will incur a second additional x on the other side, done rigourously [because the number of terms to be summed up isn't a constant number anymore, but varies also according to the number x]

edit: confirmed in wikipedia
http://en.wikipedia.org/wiki/Invalid_proof..._that_2_.3D_1_2

This post has been edited by bgeh: Jul 10 2009, 01:05 PM

thinkingfox: No, that isn't my answer, because I can't follow his steps to arrive at that solution

lin00b: I'll give it some thought and see if I can arrive at the same answer through another method, but yes some variation of the product rule will probably be needed here (which seems to be what the wikipedia picture thinkingfox pasted put on his post)

But what happens if y = x? That's the interesting part

Then you're claiming that it's useless to even try to differentiate anything more than a linear function, which is quite a dangerous proposition, no?

[and no, it's not only from graphs - graphs are useful because they give us an intuitive idea of what differentiation does]

This post has been edited by bgeh: Jul 16 2009, 06:11 PM

Okay, let's examine your claims:

You've gone on to consider another problem, which is the case where the variable we differentiate wrt (x) is different (in this case, you denoted it by y)

But what relevance does it have compared to the original problem at hand firstly?

But then the question would be: How would you differentiate x^2 then, if y can never be equal to x? The thing should give consistent results no matter what representation you use. If one representation gives you a result but the other doesn't you're in trouble!

But why can't y be equal to x in the original question? That's the question I'm asking. They're different representations of the same variable, and should give consistent results, and unless you can show why such a representation isn't allowed when considering differentiation, your claim will lead to many many problems, as described above.

This post has been edited by bgeh: Jul 17 2009, 12:55 AM

No, it's different representations of the same variable. Clearly we are agreed that 1 + 1 + 1 + 1 = 2*2 = 2 + 2 = 4. They're all exactly the same by equality [well, to be exact, the relation is transitive], but they're multiple representations of the same value.

Or equivalently, why do they have to be different variables?

This post has been edited by bgeh: Jul 17 2009, 01:35 AM

MattL: It's a rhetorical question directed at loongchai, so perhaps you should try explaining it to him instead of me

Pardon the nitpicking:

Actually, that isn't the 'true' definition of the derivative. You've provided the notion of what the derivative is, namely that it is the rate of change of some function with respect to some variable at some certain value of that variable.

Also, the '...' was actually defined. It had already been stated that the sum occurs x times (yes it's defined, even if x is not a positive natural number).

But otherwise you're mostly right, nailing down that the problem is we've not considered the fact that the x's are summed x times, which is in itself a variable

What you quoted agrees with what I said, that informally, or loosely speaking, the derivative is the rate of change of some function, or equivalently, the notion of the derivative. It is not, however, the 'true' definition of a derivative, which is what you stated. You'd need limits to construct the definition of the derivative.

This post has been edited by bgeh: Aug 1 2009, 04:52 PM