Cryptic

We know that the derivative of x2 with respect to x is 2x. However, what if we rewrite x2 as the sum of x x's, and then take the derivative:

d/dx[ x2 ] = d/dx[ x + x + x + ... (x times) ]

= d/dx[x] + d/dx[x] + d/dx[x] ... (x times)

= 1 + 1 + 1 + ... (x times)

= x


This argument shows that the derivative of x2 with respect to x is actually x. So what's going on here?

Note: Most people with some math experience can show that some part of the argument is erroneous. As in simply, something doesn't follow. However, a full solution will explain why this argument attacks something that lies at the very heart of calculus itself, and that is what really explains why it's erroneous.

phukraut

The way you've written x^2, you have defined x to be a positive integer, in fact:

x^2 = \sum_{n=1}^x (x).

Thus, x^2 is not, according to this definition, continuous. When you are differentiating, you are in fact differentiating the x that is summed over, but you're not touching the x in the index, thus the answers don't match up.

Cryptic

I waited an hour for someone to come up with that answer.

__________________
F=MA
2.998*108ms-1
Ek = 1/2mv2
R D R R

phukraut

dude I'm sorry, I just saw this now. Next time send me a PM when you need a faster answer Cheers.

Cryptic

lol, will do

__________________
F=MA
2.998*108ms-1
Ek = 1/2mv2
R D R R