# Derivative of infinity

1. Mar 11, 2008

### jimmypoopins

i got into a minor argument with a buddy of mine, he said the derivative of infinity is zero, and i argued that you can't take the derivative of infinity.

my argument was that by definition of derivative there isn't a function that can equal infinity, so you can't take the derivative of it. also, even though infinity isn't a number, theoretically infinity + 1 = infinity so it's increasing, but infinity - 1 = infinity, i.e. you can't find a slope for it at any point.

his argument was that infinity is a constant, so then it is differentiable.

i believe i'm correct but i'm not formally aware as to why, and i was wondering if you guys could give me some insight.

thanks

2. Mar 11, 2008

### Dragonfall

Infinity isn't even a number. It's like saying "the derivative of chair is 0". You're right in saying that it is undefined.

3. Mar 11, 2008

### Feldoh

According to my Ti-89 infinity is treated as a sort of constant, and thus the derivative is zero.

4. Mar 11, 2008

### MasterTinker

Infinity is undefined

To be honest the 'derivative' of infinity is very likely to be undefined and it should be of no concern as it has no practical nor theoretical purpose. If you just want a "what if it happened to be practical" answer, anything differentiable must be continuous. If you think of a function y(x) with a vertical asymptote at x = 0 (for example) where lim(x -> 0+) y(x) = (infinity) and lim(x -> 0-) y(x) = (infinity), the function is considered discontinuous at x = 0 because of the infinite limits and is therefore not differentiable (as differentiability requires continuity). This is very informal reasoning but you can see that if at any point in a function (even one defined to be 'infinity', which I'm pretty sure you cannot do) infinity is reached, then there is a discontinuity and therefore no differentiability.

5. Mar 11, 2008

### jimmypoopins

thanks for the replies, i think that's a sufficient enough answer for both of us.

yeah, i know it's not practical at all but the question was bothering both of us, even if it wasn't practical at all.

6. Mar 11, 2008

### MasterTinker

Okay, sorry for being blunt, I think it's good that you were interested.