Is the Infinite Derivative of a Function Equal to Zero?

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In summary, the conversation discusses the concept of taking the derivative of infinity, but concludes that it is a meaningless expression since infinity is a limit, not a function. The conversation also explores the idea of taking an infinite number of derivatives and the result varies depending on the function. Ultimately, the conversation concludes that there is no definitive answer to the question and it may be best to avoid dealing with the concept of infinity in derivatives.
  • #1
Dauden
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I was just thinking about this earlier today...

Is the derivative of infinity equal to zero?

Then the more interesting question, is there such thing as an infinite derivative? (I battled with Latex for 20 minutes and can't figure out how to work it so bare with me)

d^inf/dx^inf

And assuming it does exist and that (d^inf/fx^inf) of any x^p is zero where p is a positive integer. Would the infinite derivative of x^inf equal zero?

Has anyone else come across anything like this or am I just assuming too much about the derivative itself?
 
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  • #2
Derivatives are rates of change. Taking the derivative of infinity is simply non-sense since infinity is a limit, not a function.
 
  • #3
Your language is confused. Are you taking the derivative of infinity (in which case we need a definition of "infinity" as a function), or are you taking the derivative an infinite number of times as suggested by your notation d^inf/fx^inf?

For the latter, I can interpret it thus: define [tex]\frac{d^\infty}{dx^\infty}f(x)=\lim_{n\to\infty}\frac{d^n}{dx^n}f(x)[/tex]. With that definition, [tex]\frac{d^\infty}{dx^\infty}x^7=0[/tex] but [tex]\frac{d^\infty}{dx^\infty}e^x=e^x[/tex].

For the former, if you define [tex]\infty(x)=\infty[/tex] (on, say, the Riemann sphere) then [tex]\infty'(x)=0[/tex].

As to your mention of [tex]x^\infty[/tex], I don't know what that means. Formally, I could write [tex]\frac{d^\infty}{dx^\infty}x^\infty=\infty![/tex] but that doesn't have meaning in any structure I can think of.
 
  • #4
CRGreathouse said:
As to your mention of [tex]x^\infty[/tex], I don't know what that means. Formally, I could write [tex]\frac{d^\infty}{dx^\infty}x^\infty=\infty![/tex] but that doesn't have meaning in any structure I can think of.

I disagree with your meaningless expression, CRGreathouse.

To me, the proper meaningless expression should be:
[tex]\frac{d^\infty}{dx^\infty}x^\infty=\infty^{\infty}x^{\infty}[/tex]
:smile:
 
  • #5
I still find CRGreathouse's meaningless expression more mathematically sound :biggrin:

Dauden, it's like asking what [tex]\infty -\infty[/tex] is equal to. It could be zero, it could be finite, or it could be [itex]\pm \infty[/itex]. In other words, since we don't know the answer to this, we can't tell you what the answer to the infinite derivative of a polynomial of infinite degree is.
 
  • #6
arildno said:
I disagree with your meaningless expression, CRGreathouse.

To me, the proper meaningless expression should be:
[tex]\frac{d^\infty}{dx^\infty}x^\infty=\infty^{\infty}x^{\infty}[/tex]
:smile:

That interpretation, like 0, is every bit as meaningful as my result. :uhh: :biggrin:
 
  • #7
Mentallic said:
I still find CRGreathouse's meaningless expression more mathematically sound :biggrin:

:rofl:

Yes. the sweet, sickly odour of a decomposing, dead corpse is still clinging to his expression; mine is utterly desiccated, and hence more pleasant to hang around with...
 

1. What is a derivative?

A derivative is a mathematical concept that represents the rate of change of one variable with respect to another variable. It is often used to calculate instantaneous rates of change in functions.

2. How do you find the derivative of a function?

To find the derivative of a function, you can use the rules of differentiation, such as the power rule, product rule, quotient rule, or chain rule. These rules allow you to find the derivative of a function by manipulating its algebraic expression.

3. What is the purpose of finding derivatives?

The purpose of finding derivatives is to understand how a function changes over time, or the rate of change of a function. This can be useful in many areas of science, such as physics, economics, and engineering.

4. What is an odd derivative?

An odd derivative refers to the derivative of a function that is odd, meaning that it is symmetric about the origin. This means that the function has rotational symmetry of 180 degrees.

5. How do you graph an odd derivative?

To graph an odd derivative, you can use the knowledge that it is symmetric about the origin. This means that you can plot points on one side of the origin and then reflect them across the origin to get the full graph. You can also use the rules of differentiation to find the slope of the function at different points, which can help you accurately graph the derivative.

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