As we know that the derivative of 
i.e.,
However, suppose we write
i.e.,
Now let
then,
This argument appears to show that the derivative of
Where is the error?
Well!
I got following faults:
•One should note that
But if we sum negatives any number of times, we get a negative number only.
Thus, x²=x+x+….x (x times) [this expression is proven wrong below] or 
•The number of x’s being added is not a constant. Not only is x changing; the number of x’s is also changing. And one must note that nx=x+x+x+…..+x (x written n times)
But as we know that nx=xn
Thus one may think that xn=n+n+n+…..+n (n written x times).
One who thinks the second above, is wrong. A number / variable added a variable (number) of times doesn’t follow the expression xn=n+n+….+n (x times) but it follows xn=x+x+….+x (n times). I mean to say that a variable/constant can’t be added a variable number of times.
Updates:
I’m a student and I wrote – what I knew. So there were many incompletenesses in my post. I got these excellent suggestions and updates on this post:
Upadate-I
where W(x) is the whole number part of x and F(x) is the fractional part of x.
So
x² = xW(x)+xF(x)
d/dx [x²]= d/dx[xW(x) +x F (x)]
(differentiation by part)
= x’•W(x)+x • W’(x) + x’•F(x) + x•F’(x)
since x’ =d/dxx=1 and with W(x) being a constant
d/dx[x²]
= W(x)+F(x)+x {W’(x)+F’(x)} As, W(x)+F(x)=x & W’(x)+F’(x)=1 so d/dxx² = x+x=2x
[Credit: goofindoo]
Update-II
Yesmanapple sent his view on this article. Have a look.
Update-III
wnoise suggested this link:
Update-IV
This may be an excellent additional reading for readers, which was published on one of the my favorite blogs – Republic of Mathematics:
Practical Applied Mathematics: Wherein Multiplication is treated as repeated addition
updated: 02/14/2011 11:38 am.
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- A blog by Pavel Penev – On my quest for mastery (pavelpenev.posterous.com)
You simply failed to take account of the fact that not only the value of x changes, but also the size of the set itself, which you didn’t. In reaction to the second reply:
x² = xW(x)+xF(x) Why not just write x² = xW(x) = x*x ? Then you can differentiate this by parts as well.
And why isn’t multiplication repeated addition? The blog only says it isn’t, without explaining why. As far as I know, multiplication is repeated addition. This fact is very useful if you need to multiply long numbers, like 1,345,843 *3,464,901, in your head or with paper.
Hi! Thanks for your comment.
is true, if and only if x is a positive integer.
But x*x is as same as:
x*x =x*([x]+{x})
where [x] is integer part of x and {x} is fractional part of x. This post is very old and it need to be edited since I had used W(x) and F(x) for [x] and {x} respectively.
Regarding, multiplication is not repeated adition: How can you explain—
, or 
as addition? One can’t add any number fractional number or negative number of times.
4^2 = 4 * 4 = 4 + 4 + 4 + 4
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