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Homework Help: Proof by induction problem

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

    Prove by induction that the [tex]n^{th}[/tex] derivative of f(x)=[tex]\sqrt{1-x}[/tex] is

    [tex]f^{(n)}(x)=-\frac{(2n)!}{4^{n}n!(2n-1)}(1-x)^{\frac{1}{2}-n}[/tex]

    for all n geater or equal to 1

    2. Relevant equations



    3. The attempt at a solution

    To start I showed that it is true for n=1.

    Then I assumed true for all n=k. Now test for n=k+1.

    [tex]f^{(k+1)}(x)=\frac{(2k)!}{4^{k}k!(2k-1)}(1-x)^{\frac{1}{2}-k-1}(\frac{1}{2}-k)[/tex]

    From here I rearranged and multiplyed by 4/4 and (k+1)/(k+1) to obtain

    [tex]f^{(k+1)}(x)=\frac{(2k)!(2-4k)(k+1)}{4^{k+1}(k+1)!(2k-1)}(1-x)^{\frac{1}{2}-(k+1)}[/tex]

    This is where i got stuck.
    Was wondering if someone could tell me if I'm on the right track and/or point me in the right direction.

    I know I'm trying to get to

    [tex]f^{(k+1)}(x)=-\frac{(2(k+1))!}{4^{k+1}(k+1)!(2(k+1)-1)}(1-x)^{\frac{1}{2}-(k+1)}[/tex]

    but cant quite make the leap to get there. Any help/advise would be appreciated.
    Thanks
     
    Last edited: Oct 1, 2008
  2. jcsd
  3. Oct 1, 2008 #2

    Redbelly98

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    Staff Emeritus
    Science Advisor
    Homework Helper

    You are on the right track.

    First, just fixing a couple of typos (for the benefit of other helpers):
    You multiplied by 4/4, not 2/2
    "(2-4n)" should be (2-4k)

    Next, a couple of observations:
    What is the derivative of (1-x)m? What factor did you miss when you took the derivative before? If you're stumped by that question, I'll instead ask: what is the derivative of (1-x)?

    What extra terms are needed for (2k)! to become (2(k+1))! ?
     
  4. Oct 2, 2008 #3
    Cool thanks I got it now.
     
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