one of my friend pointed this out. admittedly it confuses the heck out of me.

2 * 2 = 2+2;
2 * 3 = 2+2+2;
2 * 4 = 2+2+2+2;
...
x * x = x+x+x+....+x+x (x number of times)

x * x =x^2
therefore,
x^2=x+x+x+x+...+x+x (x number of times)
differentiate,
d/dx(x^2)=2x
d/dx(x+x+x+...+x+x)=1+1+1+1+...+1+1+1 (x number of times)
which equals to x

therefore,
2x=x,
2=1


This post has been edited by lin00b: Jul 10 2009, 01:01 PM

mana question??

i think ur logic is flawed

I'm not really any good in maths and very rusty in calculus already, but why would d(x^2)/dx equals to 2x? Shouldn't it be just x?

please point out the flaw

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Added on July 10, 2009, 12:55 pmno question (ok, maybe i mis-named the topic). sorry
just a solution?

This post has been edited by lin00b: Jul 10 2009, 12:55 PM

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

numbers never lie to us...


This post has been edited by vivienne85: Jul 10 2009, 01:44 PM

Source: Wikipedia: Invalid Proof

This post has been edited by Thinkingfox: Jul 15 2009, 09:00 PM

I'm following right up to
2x=(1+1+1+...+1)(x+x+x+...+x)

then its again

isnt 2x= x + x ?

ya...

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)

What happen to students now days

Never seem understand the fundamental of function x^2 and 2x.

Plot the graph TS and find the real meaning of differentiate w.r.t (x)

Why /k/ always think it is calculus every time there is d()/dx ???

Don't tell me TS is a uni student

These math tricks are not uncommon. No harm in looking at it in thinking over it.

And what does /k/ has to do with this

d/dx(x^2)=2x
d/dx(x+x+x+...+x+x)=1+1+1+1+...+1+1+1 (2x number of times)
which equals to 2x

therefore,
2x=2x,
2=2

why not like this??

2 * 2 = 2+2;
2 * 3 = 2+2+2;
2 * 4 = 2+2+2+2;
...
x * y = x+x+x+....+x+x (y number of times)

since when did x become a constant??? <removed>.jpg

sorry then, lets make it this way:
2 * 2 = 2+2
3 * 3 = 3+3+3
4 * 4 = 4+4+4+4
x * x =x+x+x+...+x (x number of times)

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Added on July 15, 2009, 6:08 pmhow does d[x+x+x+..+x (x times)]/dx = 1+1+1+...+1 (2x times)?

This post has been edited by lin00b: Jul 15 2009, 06:08 PM

Oops.. I'm sorry about that.

d/dx(x^2)=2x
d/dx(x+x+x+...+x+x)=1+1+1+1+...+1+1+1 (2x number of times)

2x because of d/dx(x^2)=2x

d/dx(x^2) = d/dx(x+x+x+...+x+x)

and

2x = 1+1+1+1+...+1+1+1 (2x number of times)

d/dx(x) = 1
d/dx(x+x) = 1+1
d/dx(x+x+x)=1+1+1
...
d/dx(x+x+x+...+x [x number of times])=1+1+1+...+1 [x number of times]

no?

u should look at it this way:

d/dx(x) = 1
d/dx(x+x) = 1+1
d/dx(x+x+x)=1+1+1
...
d/dx(x+x+x+...+x [y number of times])=1+1+1+...+1 [y number of times]

d/dx(yx)=1+1+1+...+1 [y number of times]
d/dx(yx)=y

use y as a factor instead of x, so it won't affect the differentiation in terms of x.

I guess, to really understand this, u gotta search deep into the Differentiation from First Principle

This post has been edited by loongchai: Jul 16 2009, 01:49 AM

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

y is a factor.

y cannot be x because it causes the line of the graph to be different.

Differentiation is derived from graphs.


For example,

From the straight line equation, y=mx+c,

dy/dx is the slope of the graph or m.

If you were to replace m with x, the graph would not be linear anymore. (y = x^2 + c) You would be changing the linear line to a curve.

In this case, m is a factor. Same goes to y from the TS's question.

I am not very good at explaining.

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

No, of course we can differentiate more than a linear graph. If you were to add a x factor to a linear function, it turns to a curve. And if you were to add another x to the curve function, it becomes another type of graph. And it goes on. However, adding a factor instead of x would only change the scale of the graph.

This post has been edited by loongchai: Jul 16 2009, 06:15 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?

The original problem will only happen if we were to equate y to x or y=x. I am trying to prove that y cannot be equal to x. This is because if y=x, then the graph will be different, which affects the differentiation. y, being a factor will only affect the scale of the graph, not the shape of it.

Wah this discussion still going on arr... See the wiki site on the previous page la... While it's true that
x^2=x+x+x+x+...+x+x (x number of times)
and
d/dx(x^2)=2x,

the mistake is
d/dx(x+x+x+...+x+x)=1+1+1+1+...+1+1+1 (x number of times)
because x here is not held constant.

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!

You are making things complicated...

y can never be equal to x? That only applies to the original question.



This post has been edited by loongchai: Jul 16 2009, 11:25 PM

Erm, okay...I would just wanna say...

Just think this
Apples and Oranges are all put together. How can u use x to represent the number of apples, and ALSO to represent the number of oranges? You have to use another variable, for example 'y', to represent the oranges.

no matter wat happen, y cannot be x. they can be equal in figures, but not the things they are representing. now apply this back into the things u 2 are debating over. =)


^^

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

yes. u said it, DIFFERENT REPRESENTATIONS, but NOT the same VARIABLE. they just have the same FIGURE.

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

dunno lol..al i noe..a,b,c r variable..hehe

Of course you can differentiate x^2 and y can equal x. Just that when you make it equal to x, you have to differentiate it as well now.
d/dx(y*x) = d/dx(y)*x + y*d/dx(x) = 0+y = y
and
d/dx(x*x) = d/dx(x)*x + x*d/dx(x) = x+x = 2x

If you follow the example in the first post, you'll be differentiating only one x while there are actually two there. Which means you don't apply the multiplication law.

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

d/dx (x+x+...x+x) is in fact flawed. Remember that x increase at the magnitude of power of 2;
Let say x = 10
d/dx(x^2) = x+x+...x+x (10 times, with the value of x = 10)
Let say x = 11
d/dx(x^2) = x+x+...x+x (11 times, with the value of x = 11)
Why I say it is flawed, because the "x number of times" is not reflected in the equation, but only the value of x.

The true definition of d/dx is rate of change. For the equation y=x^2, the derivative (d/dx=2x) simply means the rate of change for the function when x is at certain value.

Let x = 10;
d/dx (x^2) = 2x = 20;

By rate of change, it means that any changes to the value, it is effectively scaled with that magnitude.

Say x = 10.001; the change (dx) = 0.001;
Effectively, the result should be = 10 + 0.001*20 = 10.02
Double check 10.001^2 = 10.020001

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Added on July 28, 2009, 11:56 pmNo offense, but this question cannot be solved with only the understanding of secondary add. math. What was thaught in secondary add. math class is just the foundation. They never teach about the principle of derivative and integration (calculus). Once you start reading engineering maths and other tertiary maths, especially on application, then only you can see the whole picture. For now, just follow the rules. Don't act smart and go change the equation.

Equation is just a function that relate the variable. And in real world application, function is a combination of derivative and integration variable.

This post has been edited by Aurora: Jul 28 2009, 11:58 PM

true, the explanation is there, but how many actually understand the explanation?

and whats the problem with x not being held constant? is any mathematical laws being broken by letting x remain a variable?

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Added on July 30, 2009, 1:39 pmlet x = 10,
2x = 1+1+1+... (10 times)
20 = 10

iirc, small changes is only used when there is 2 or more variable in a function? (could be wrong though)

This post has been edited by lin00b: Jul 30 2009, 01:39 PM

Yes. That is the true definition of derivative.

20 = 10(?)
You can't compare like that la, bro. f(x) = (x*x) = x+x+...x+x is flawed the "...." was not define. Also, does that mean:
d/dx (x^3) = d/dx(x+x+....x+x) (x^2 number of times)
d/dx (x^3) = 1+1+....1+1

if x=10;
d/dx (x^3) = 1+1+....1+1 (10^2 number of times)
d/dx (x^3) = 100

but, d/dx(x^3) = 3x^2 = 300

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

From wikipedia:


In school, to avoid confusion, it is often express as the slope of a graph. In fact, some math teachers could hardly appreciate the rate of change. The development of calculus, derivative and integration was largely due to extensive study of transient (the rate of change and behavior) of different physic aspect.

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