x+x+x+x+x+...... Up to finite x times= x^2
Differentiating both sides we get 1+1+1+1+1......=2x What is wrong in this?? Attempt: I'm thinking that the above statement written is only for natural numbers..
x+x+x+x+x+...... Up to finite x times= x^2
Differentiating both sides we get 1+1+1+1+1......=2x What is wrong in this?? Attempt: I'm thinking that the above statement written is only for natural numbers..
I've seen this question before, but apparently not on math.SE. I'm a bit surprised it's not a duplicate; it's quite a common thing to come up with.
The trick is in the sneaky "…" you included. How does one differentiate "…"?
Consider instead the very similar question:
$1+1+\dots+1 = x$. So if we differentiate both sides, we should get $0+0+\dots+0$ on the LHS, and $1$ on the RHS. What have I done wrong?
The answer is that you've tried to differentiate a sum whose length varies with $x$. To do this properly, you're going to have to use the chain rule. Linearity of differentiation only works when the length of the sum is fixed in advance.