Just a little brush up on skills

[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.

[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