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Second Derivative of Exponential Function

  1. Feb 23, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the second derivative of:

    [tex] e^{ax} [/tex]

    and

    [tex] e^{-ax} [/tex]

    2. Relevant equations



    3. The attempt at a solution

    The book that I am using seems to have been very vague on how to take the derivatives of exponential functions. I am aware that:

    [tex] \frac {d(e^{x})} {dx} = e^{x} [/tex]

    but it says literally nothing about how the chain rule applies to exponential function, or does it? Am I just making it too difficult? Please help!
     
  2. jcsd
  3. Feb 23, 2010 #2

    Mark44

    Staff: Mentor

    [tex] \frac {d(e^{u})} {dx} = e^{u} \frac{du}{dx} [/tex]
     
  4. Feb 23, 2010 #3
    Assuming "a" as a constant, you can consider two functions.

    One is [tex]f(y)=e^y[/tex] and the other [tex]y=ax[/tex], so the derivative in respect to x would be:

    [tex]\frac{df}{dx}=\frac{df}{dy}\frac{dy}{dx}=...[/tex].

    You just need to calculate the differentials tand to the same again to get the result.
     
    Last edited: Feb 23, 2010
  5. Feb 23, 2010 #4
    Ok so I tried the method listed, I am hoping someone can confirm that this is correct:

    [tex] y = e^{ax} [/tex]

    [tex] y' = a(e^{ax}) [/tex]

    [tex] y'' = a^2(e^{ax}) [/tex]

    Thanks!
     
  6. Feb 23, 2010 #5
    That's it ;)
     
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