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What is the derivative of u(w)=k-e^(-aw)

  1. Jul 22, 2008 #1
    1. The problem statement, all variables and given/known data

    I have two questions. First:

    What is the derivative of u(w)=k-e^(-aw)

    Second:

    For U(w)=sqrt(w), prove that U(pie(x)+(1-pie)y) > pie*U(x)+(1-pie)U(y)

    2. Relevant equations



    3. The attempt at a solution

    For the first one, I haven't seen this type of problem in so long that I just don't remember how to take a derivative in this form. Would it be:

    u'(w)= -(-aw)e^(-aw-1)*-a
    =(aw)e^(a^2w+a)?

    I wasn't sure if the chain rule was appropriate here.

    For the second question, I have:

    sqrt(pie(x)+(1-pie)y) > pie*sqrt(x)+(1-pie)sqrt(y) so...

    (pie(x)+(1-pie)y)^(1/2) > pie*(x)^1/2+(1-pie)(y)^1/2

    From here I don't know where to go. I don't have much experience with proofs so can anyone give some guidance? Thanks for your help!
     
  2. jcsd
  3. Jul 22, 2008 #2

    Dick

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    Re: questions

    e^(aw) is not a power law. d/dw(exp(aw)) is a*exp(aw). Try again.
     
  4. Jul 23, 2008 #3
    Re: questions

    Sorry I'm a little confused by your reply, does that mean for U(w)=k-e^(-aw) that
    U'(w)=(k-e^(-aw))' =-a(-e)^(-aw) =ae^(-aw)? Thanks again for your help.
     
  5. Jul 23, 2008 #4

    Dick

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    Re: questions

    The answer is right. -a(-e)^(-aw) is a bit confusing. You might want to move a parenthesis in there.
     
  6. Jul 23, 2008 #5
    Re: questions

    hmm I see the notation is a bit weird. But the answer comes out correct if written as
    -a(-e^(-aw)) right? Thanks again.
     
  7. Jul 23, 2008 #6

    Dick

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    Re: questions

    Sure. I just didn't like (-e)^(-aw).
     
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