Differentiating infinity wrt any variable: is it zero or undefined?

Question

I want to further the discission from this question: Can you take the derivative of a function at infinity?

Basically I am interesting in this expression: $\frac{d}{dx}(\infty) \text{ or } \frac{d}{dx}(-\infty)$

Are those expressions zero or undefined? Software like TI-89 calculator and Mathematica will treat infinity as a constant so the expressions become zero (and hance defined).

My motivation for asking: IF the expressions equal zero and is defined then working with improper integrals will become much simpler by use of for instance Leibniz rule for integration.

The answer to this question How do I differentiate an improper integral? might imply that differentiating infinity with respect to x is zero.

This is undefined. Derivatives are defined for functions, not for anything you want. This question is the same as asking "what is $\frac{d}{dx} (\mathrm{Hello!})$?" — Crostul, Sep 30 '18 at 18:07

score 4 · Answer 1 · answered Sep 30 '18 at 19:09

4

Infinity is not a number. $\infty x$ Is not defined in basically every context. The basic definition of a derivative does not deal with things that are not defined.

answered Sep 30 '18 at 19:09

clathratus

17,161

Differentiating infinity wrt any variable: is it zero or undefined?

1 Answers1