Can a Function Equal Its Own Derivative?

  • Context: Undergrad 
  • Thread starter Thread starter Axecutioner
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Discussion Overview

The discussion centers around the question of whether a function can be equal to its own derivative, exploring both theoretical and mathematical aspects of this concept.

Discussion Character

  • Exploratory, Mathematical reasoning

Main Points Raised

  • One participant questions the possibility of a function f(x) being equal to its derivative f '(x).
  • Another participant suggests that the exponential function f(x) = e^x satisfies this condition, as its derivative is also e^x.
  • A later reply mentions that solving the differential equation f'(x) = f(x) leads to the conclusion that any multiple of e^x will satisfy the condition.
  • Another participant confirms that functions of the form f(x) = Ce^x, where C is a real number, are solutions, asserting that these are the only functions that equal their own derivative.

Areas of Agreement / Disagreement

While there is agreement on the exponential function and its multiples being solutions, the discussion does not explore or confirm the existence of other potential functions that could meet the criteria, leaving the topic open for further exploration.

Contextual Notes

The discussion does not address potential limitations or assumptions regarding the types of functions considered, nor does it explore the implications of the solutions provided.

Who May Find This Useful

Readers interested in differential equations, calculus, or the properties of exponential functions may find this discussion relevant.

Axecutioner
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Is it possible for f(x) to be equal to f '(x)? I've been trying all day to make it work and have gotten nothing.

Thanks
 
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Axecutioner said:
Is it possible for f(x) to be equal to f '(x)? I've been trying all day to make it work and have gotten nothing.

Thanks

Sure, if f(x)=ex, then f'(x) = ex = f(x).
 
Last edited by a moderator:
Oh, didn't think about that. XD I kept trying Trig to make it work...

Are there any other ways?
 
Axecutioner said:
Oh, didn't think about that. XD I kept trying Trig to make it work...

Are there any other ways?

if you solve the differential equation

f'(x)=f(x)

You'd see that any multiply of ex would satisfy the condition. I think that is the only function, that I know of anyhow.
 
Indeed, every function of the form [tex]f(x)=Ce^x[/tex], with [tex]C\in \mathbb{R}[/tex] will do. There are no other functions that equal it's own derivative...
 

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