Why is this wrong?

AvgStudent

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x+x+x+...+xx times=xx=x2\underbrace{x+x+x+...+x}_\text{x times}=x*x=x^2Differentiate both sides:
dxdx+dxdx+dxdx+...+dxdxx times=2x\underbrace{\frac{dx}{dx}+\frac{dx}{dx}+\frac{dx}{dx}+...+\frac{dx}{dx}}_{\text{x times}}=2x1+1+1+...+1x times=2x\underbrace{1+1+1+...+1}_{\text{x times}}=2xx=2xx=2x1=21=2
 
I just have a few question for you.

What do you get if you add 1 to itself 7 1/2 times?
What do you get if you add 1 to itself 2.92 times?
What do you get if you add 1 to itself sqrt(2) times?
What do you get if you add 1 to itself -5 times?
and my favorite one--What do you get if you add 1 to itself pi times?
 
I try to avoid π\pi. It'll go on and on forever 8-)
Why will it go on forever? How about sqrt(2), will it go on forever? How about 1/3?
If I asked you to add 1 to itself pi times, then it'll go on and on forever.
The only 'it' I can think of is the adding of 1's.
So if you add 1 to itself pi times, then you get 1+1 + 1 + ... = infinity?
That is strange as pi is less than 4 and if I add 1 to itself 4 times, 1+1+1+1, I get 4.
The point topsquark and I are trying to make is that you can't add 1 to itself other than a positive integer amount of times.
 
Why will it go on forever? How about sqrt(2), will it go on forever? How about 1/3?
If I asked you to add 1 to itself pi times, then it'll go on and on forever.
The only 'it' I can think of is the adding of 1's.
So if you add 1 to itself pi times, then you get 1+1 + 1 + ... = infinity?
That is strange as pi is less than 4 and if I add 1 to itself 4 times, 1+1+1+1, I get 4.
The point topsquark and I are trying to make is that you can't add 1 to itself other than a positive integer amount of times.
I think he/she got the point. It was a pi joke.
 
I think the main problem here is that the "x times" notation is really an obfuscated, hidden, operator that is being missed out in the differentiation process.

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If this is changed to a constant "c times" then it works...

x+x+x+...+xc times=cx\underbrace{x+x+x+...+x}_\text{c times}=c*x
dxdx+dxdx+dxdx+...+dxdxc times=c\underbrace{\frac{dx}{dx}+\frac{dx}{dx}+\frac{dx}{dx}+...+\frac{dx}{dx}}_{\text{c times}}=c
1+1+1+...+1c times=c\underbrace{1+1+1+...+1}_{\text{c times}}=c
c=cc=c
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If we make the operator more explicit then it works...

x+x+x+...+xx times=x2\underbrace{x+x+x+...+x}_\text{x times}=x^2
xx=x2x*x=x^2
Differentiate (product rule on the LHS):

xdxdx+dxdxx=2xx*\frac{dx}{dx}+\frac{dx}{dx}*x=2x
2x=2x2x=2x
 
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