This a tricky math question I encountered. I know a little bit about the answer. But I want somebody who is very good at math to help me find the real reason behind this.
OK
Lets start
$1^2 = 1$
$2^2 = 2+2$
$3^2 = 3+3+3$
...........................
................................
$x^2 = x+x+x+x+.....(x times)$
Differentiating with respect to $x$
We get
$2x = 1+1+1+1...... (x times)$
which is equal to
$2x = 1*x$
$2x = x$
Which is incorrect.
Where did I go wrong?? :O :O
Please think of the case when $x$ belongs to a real number and not an integer.
Then you will get the idea, say what happens when x = 1.5?
$ x^2 = x * x $
$\frac{d}{dx} x^2 = x + x = 2x $
Your problem arises because you have made the assumption that $$\frac{d}{dx}\sum_{i=1}^{x}x$$ can be calculated by simply summing the derivatives of the individual summands; i.e., you have made the assumption that $$\frac{d}{dx}\sum_{i=1}^{x}x=\sum_{i=1}^{x}\frac{d}{dx}x.$$ This is not true; the value of $x$ affects both the elements of the sum and the number of elements in the sum.
