i get it and i don't get it...

I know that. (And, obviously everyone else does too.)
Did you go through the next line : D(nx) = nD(x) ONLY for constant value of n.

You can say D(x + x + x) = 3.
But, D(x + x + ..(y times).. + x) != y when y depends on x.

When y depends on x, D(x + x + ..(y times).. + x) = yD(x) + xD(y).
You have to apply product formula 'coz you are expressing one function (in this case, x) as the sum with itself, another function (in this case, y) times.
In your question, y depends on x as y=x.
So, f'(x) = xD(x) + xD(x) = 2x != x.

And, LOL @ what you mentioned as the correct answer.
You are trying to show that 2x != x by substituting x = 4 !!

The first principle and blah.. blah.. blah.. was simply unnecessary. Yeah it was unnecessary. You get the same results by differentiation formula too and you can still show x != 2x by showing 4 != 8. LOL

I wonder the basis on which my answer was rejected. The correct answer (by the OP) sounds too silly.

And, one more thing; the following (in bold) is the silliest thing I have ever seen in my life :
If any one on the earth would accept that, I am ready to leave programming and maths the very day.

Could you give me a single value for N where that doesn't hold please?
(By construction this must be a positive integer, of course)

As I gave two examples where it does hold I think that's fair enough.
Let me repeat those two examples just in case it's not clear:

Let's say x=3. f(x)=x+x+x; f'(x)=1+1+1=3 so f(x)=x.
x=7: f(x)=x+x+x+x+x+x+x; f'(x)=1+1+1+1+1+1+1=7 so f(x)=x.

So let's see a counter-example rather than just a declaration that I'm silly.

Correction (missed the '):

Let's say x=3. f(x)=x+x+x; f'(x)=1+1+1=3 so f'(x)=x.
x=7: f(x)=x+x+x+x+x+x+x; f'(x)=1+1+1+1+1+1+1=7 so f'(x)=x.

this is gonna be controversial....

It doesn't have to be. If Saswat stops being confrontational and sticks to the facts without bothering with the insults then this can be sorted out very easily.

I was offline 'coz there was some problem with the broadband connection here.
Things have been sorted out now ....

Let's get back to the topic then.

Just go to some maths prof, and if he agrees that f'(x) = x is true, I will truly give up maths and programming. Just announce it here that the prof agrees that f'(x) = x.

And, what counter-example are you talking abt ?
Differentiation is not a game or something -- calculus is a vast branch of Maths and what you are trying to show will prove Newton and Leibniz's work false !!

As I said initially -- I am reiterating, taking x = 3, or x = 7 is a very big misconception of yours.
Let me talk your way :

Let's have x = 3.
So, f(x) = x + x + x = 3 + 3 + 3 = 9
=> f'(x) = 0.
Agree ??

Same for x= 7 or anything. Now, do you agree to this ???


Please do seriously revise differentiation, b4 next post. And, I am serious abt this thing, not 'coz it's some contest question or anything; but 'coz I love maths.

And, I do stick to facts ... but not false ones.
And I can very boldly and (may be rudely) oppose those who support any false facts.

OK, let's take N items instead of x items.
Let's set N=5 and so f(x)=x+x+x+x+x - that's 5 X's, because N=5.
Let's also completely ignore the value of x for now.

Now let's differentiate f(x); this gives 1+1+1+1+1, which is 5.
We can further simplify f(x)=5x and this differentiates directly to 5, because d/dx (ax^b) = abx^(b-1).

Do you at least agree with that?

Taking N = 5 instead of x is OK,
'coz N is independent of x and hence can be treated as a constant.

Do, D(Nx) = ND(x).

But you can't do so when N depends on x.
Like, you can't take N = x.