# incorrect diffrentiation

#### recca02

Joined Apr 2, 2007
1,212
this is an old trick at least for the biggies here but please care to elaborate on the below example

1+1+1......n times=n
1+1+1... x times = x ...x is a variable -----(1)
x+x+x....n times = x.n
x+x+x....x times = x^2
diff w.r.t x
1+1+1... x = 2.x
from (1)
x = 2.x
or
1 = 2
i know diff isnt allowed since limit does not exist in the case of
x+x+x....x times
but somehow i want a little detailed explanation
wud highly appreciate it if someone can help.

#### Dave

Joined Nov 17, 2003
6,969
1+1+1... x times = x ...x is a variable -----(1)
diff w.r.t x
1+1+1... x = 2.x
d/dx(x) = 1 not 2.x

i.e. the rate of change of x is 1, which from 1+1+1... x is clearly true.

Dave

#### recca02

Joined Apr 2, 2007
1,212
but d/dx(x^2) = is 2.x (that is what i intended)
i mean diff both rhs and lhs
there is nothin wrong with diff the only thing is it is not allowed in this case.

#### DrNick

Joined Dec 13, 2006
110
I think your problem is that you are using a dummy variable as part of the limit in your sum, making the summation non-linear. So you have
sum(x, i=0, x)

if we differentiate both sides you will notice that since the summation is no longer linear we cant interchange the order of summation and differentiation. SO,

d/dx ( sum(x, i=0, x) ) = d/dx(x^2) != [ sum(d/dx(x), i=0, x) = sum(1, i=0, x) ]

that x that is the upper limit must also be considered when differentiating (this is why we can't bring the derivative inside of the sum).

#### Dave

Joined Nov 17, 2003
6,969
but d/dx(x^2) = is 2.x (that is what i intended)
i mean diff both rhs and lhs
there is nothin wrong with diff the only thing is it is not allowed in this case.
Apologies I misinterpreted your initial question. I concur with the answer offered by DrNick.

Dave

#### recca02

Joined Apr 2, 2007
1,212
thanks mr. dave and drnick
anyway can anyone come up with the graph of such function
i mean x+x+x+.....x times
wud it be a step funtion with a linear component

#### Dave

Joined Nov 17, 2003
6,969
thanks mr. dave and drnick
anyway can anyone come up with the graph of such function
i mean x+x+x+.....x times
wud it be a step funtion with a linear component
Plotting (sum of x) against (i = 1 to x) gives a straight line of gradient x. Try it in Excel with x arbitrarily set to say 10.

Dave

#### recca02

Joined Apr 2, 2007
1,212
i'll bother u just once more.
how much sense does it make that a variable x which is a real number lets say
1.786 to be added 1.786 times ( X+X+..X times)
its been years since i learned summation but if i m not wrong it adds only
whole numbers it does not take into account fractional values.

#### Dave

Joined Nov 17, 2003
6,969
i'll bother u just once more.
how much sense does it make that a variable x which is a real number lets say
1.786 to be added 1.786 times ( X+X+..X times)
its been years since i learned summation but if i m not wrong it adds only
whole numbers it does not take into account fractional values.
Although it sounds wierd, 1.786 to be added 1.786 times is the same as 1.786^2 - think of it in terms of integers for a proof, and think that the rules apply equally to all real numbers.

Dave