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Does such a function as f(x)=f'(x) exist?

  1. Mar 20, 2013 #1
    i was wondering if this function actually exist or dosent if it does can someone provide an example
  2. jcsd
  3. Mar 20, 2013 #2
    Have you tried f(x) = 0 ? :)
    More seriously, there is a well-known function with this property: [itex]f(x) = e^x[/itex] .
  4. Mar 20, 2013 #3
    And to show yourself it, try solving the differential equation ##\frac{dy}{dx} = y##

    It doesn't really prove much, but its a nice way to convince yourself that what slider142 said is true!
  5. Mar 20, 2013 #4
    Or more general: ##f(x)=Ce^x## for an arbitrary constant ##C##. These are all the functions that satisfy your equation.
  6. Mar 21, 2013 #5
    Thanks we haven't gotten to logarithms yet
  7. Mar 21, 2013 #6


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    Then why did you ask this question?
  8. Mar 21, 2013 #7


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    it seems odd, pedagogically, that one would be learning about differential equations and calculus without first learning about logarithms.
  9. Mar 21, 2013 #8
    My guess is that the question arose out of OP's curiosity rather than actually being introduced to differential equations in the coursework. I remember when learning calculus I had similar curiosities myself, before even learning about exponential and logarithmic functions.

  10. Mar 25, 2013 #9


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    It sounds like the OP is in the first semester of calculus, where at some point you are eventually told that ex has the very property being discussed here. Probably the class hasn't gotten there just yet.

    Also, the OP may have meant that they haven't covered logarithms yet in the calculus course. At least that would make more sense then never having gotten to them, ever.
  11. Mar 26, 2013 #10
    It might be interesting to note that some functions are also equal to their own nth derivatives, for example:

    [itex]\frac{\mathrm{d}^2}{\mathrm{d}x^2} e^{-x} = e^{-x}[/itex]

    [itex]\frac{\mathrm{d}^3}{\mathrm{d}x^3} e^{\frac{-x}{2}}\sin(\frac{\sqrt{3}x}{2}) = e^{\frac{-x}{2}}\sin(\frac{\sqrt{3}x}{2})[/itex]

    [itex]\frac{\mathrm{d}^4}{\mathrm{d}x^4} \sin x = \sin x[/itex]
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