Reading Ideas

It's been a long time since i've dealt with derivatives, and messing around with them recently I came across a certain problem I'd had years ago.

It goes something like this :

( x^2 denotes x (raised to) 2 and so on; to clear any doubts ).

Our first equation is this :

d (x^2) = 2x   ----->  equation 1
-
dx

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Now, we know what 
2^2 = 2 + 2
3^2 = 3 + 3 + 3
4^2 = 4 + 4 + 4 + 4
... and so on

Hence in general

x^2 = x + x + x + x......x times

Take the right hand side of the equation and derivate each x.
You get 

1 + 1 + 1 + 1 + 1......x times = x ------> equation 2

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Hence from equation 1 and equation 2 we get

2x = x

Hence 

2 = 1

Heh.
Now obviously I've done something wrong here, but I cannot figure out what.

NI

x^2 = x + x + x + x......x times

Take the right hand side of the equation and derivate each x.

My guess is that you can't easily derivate a sum with an unknown number of summands.

'Yeah, That's what Jesus would do. Jesus would bomb Afghanistan. Yeah.' - snowlion

My guess is that you can't easily derivate a sum with an unknown number of summands.

More or less. You can derive sums of finitely many addends without any problems, and even infinite sums given that the derivation still converges.

However in this case the length of the sum itself is a function of x, so you'd have to derive for the length of the sum, too, which just makes no sense. (You cannot derive something discrete).

Basically, you're not really deriving x^2 but instead x*y where y is constant (and later evaluate it with x = y). The derivation of x*y is y. You're right side is y, too.

'Repent, Harlequin!' said the Ticktockman. 'Get stuffed!' the Harlequin replied, sneering.

This post was edited by wizz on Jan 09, 2004.

(You cannot derive something discrete).

From what I remember a discrete function is a function that is defined only for a set of numbers that can be listed, such as the set of whole numbers or the set of integers.

So what's wrong with derivating a discrete function?

However in this case the length of the sum itself is a function of x, so you'd have to derive for the length of the sum, too, which just makes no sense.

Maybe you're right, but can you tell me why derivating over the length of the sum would not make sense?

NI

From what I remember a discrete function is a function that is defined only for a set of numbers that can be listed, such as the set of whole numbers or the set of integers.

Discrete means that every point in the set is isolated, which means there is a small "area" around each point with no other point inside. This is not necessarily true for countable sets.
But we're thinking of the same :-)

So what's wrong with derivating a discrete function?

The derivative f'(x) can be defined as the value of the fraction (f(x)+f(x+h))/h when h approaches zero (steepness of tangent to the graph of the function in x).
With a discrete function, f(x+h) just isn't defined for small h.

Basically, in order to find a tangent the values of the function has to be known in a small neightbourhood of the point where you want to derive. As is the case for the smooth function x^2. However, your right side (x+x+...+x) just makes sense for whole values of x. It isn't defined for x = 1.01 e.g. And there's your problem.

'Repent, Harlequin!' said the Ticktockman. 'Get stuffed!' the Harlequin replied, sneering.

It isn't defined for x = 1.01 e.g. And there's your problem.

Ahh, yes. That solves the problem. Thanks.

NI

What the f*#§??? That is really weird!
Your argumentation seems to make perfect sense. But then....

Seems like you shattered the very foundation of our civilization; science, technology and everything we believed in - just rendered ad absurdum. Maybe Stephen knows the answer.

x^2 = x + x + x + x......x times Take the right hand side of the equation and derivate each x. You get 1 + 1 + 1 + 1 + 1......x times = x ------> equation 2

Mmmh, you say

x^2 = x + x + x ....

Then

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

equals

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

Then
2x = 2x.

This post was edited by deltaBird on Jan 09, 2004.

DeltaBird's answer is right...

You have to realize that 1+1+1+1+1... is <i>equal</i> to x, and so it balances

If you ignore this, you basically turn a variable into a constant while deriving it, and your answer is off

"It was not God who created man, but man who created God." --Santa Caserio