Question

my friend asked me to
find mistake..
x^2 = x+x+x+x.... x times.. ... ...
(i found one here
X^2 = (|x|)^2
= |x| + |x|+... x times.. as x^2 will always be positive, i told him to try taking x= -2)


well his ques was,
x^2 = x+x+x+x.... x times..
differentiating it wrt x
2x = 1+1+1+... x times..
2x=x
2=1 ???
o_O
are there any more mistakes..? am i right to say that..? doesnt my friend deserves a face palm for asking such questions?

Answer 1

I have seen this question before
the problem is that you are trying to differentiate a finite number of x's as if they were a constant
but x is a variable, so the number of x's in x+x+x+... is a variable
hence you cannot differentiate as if it were (x+x)'=1+1=2

Answer 2

x^2 = x+x+x+x.... x times..
...but "x times" is a variable, not a constant
it now becomes so we would be better writing d/dx(x*x)=1*x+x*1=2x
as it should be

Answer 3

@yash2651995 is this making sense to you?

Answer 4

hardly making sense to me ... LOL.

Answer 5

yes it does makes sense to me... i didn't noticed that!! thanks for your help :)

Answer 6

if we had \(x+x+x+x+x+... n\) times (where n is constant) this is the same as
\(nx\)
if we differentiate this we get \(n\)

but we have \(x+x+x+x... x\) \(times\)
so rewriting this it would be \(x(x)\), which, by the product rule, we know the derivative of is as it is supposed to be.
that is the appropriate form to take the derivative of, because that allows us to take into account the fact that the number of x's is a variable, not a constant as it was in the previous case.

Answer 7

you're welcome :)