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How to prove this equation

  1. Jan 30, 2009 #1
    i need to prove that this equation is true
    [itex]
    \frac{d^n}{dx^n}\left(x^{n-1}e^{1/x}\right) = (-1)^n\frac{e^{1/x}}{x^{n+1}}
    [/itex]

    i tried by induction
    [itex]
    f_n(x) \equiv x^{n-1}e^{1/x}
    [/itex]

    [itex]
    g_n(x) \equiv (-1)^n\frac{e^{1/x}}{x^{n+1}}
    [/itex]
    this is the pattern if i keep differentiating the base case.
    [itex]
    \frac{{d^{k + 1} }}{{dx^{k + 1} }}f_{n + 1} (x) = x\frac{{d^{k + 1} }}{{dx^{k + 1} }}f_n (x) + (k + 1)\frac{{d^k }}{{dx^k }}f_n (x)
    [/itex]

    then i tried to differentiate the "n" equation inorder to get to the n+1 equation
    [itex]
    \frac{{d^{n + 1} }}{{dx^{n + 1} }}f_n (x) = \frac{{d^{} }}{{dx^{} }}g_n (x) = ( - 1)^n {\rm{[}}e^{1/x} ( - \frac{1}{{x^2 }}{\rm{)}}x^{n + 1} + (n + 1)x^n e^{1/x} {\rm{] }} \\
    [/itex]

    but it didnt work
    ??
     
    Last edited: Jan 30, 2009
  2. jcsd
  3. Jan 30, 2009 #2

    Mark44

    Staff: Mentor

    Base case (n = 1); d/dx [itex]e^{1/x} = - e^{1/x}/x^2[/itex]
    So the equation holds for n = 1.

    Induction step (n = k)
    Asssume that [itex]d^k/dx^k(x^{k -1}e^{1/x}) = (-1)^k e^{1/x}/x^{k + 1}[/itex]

    Now show that the equation holds for n = k + 1. Note that the (k + 1)st derivative is the derivative of the kth derivative.
     
  4. Jan 30, 2009 #3
    i did that in the first post
    it doesnt give me the n+1 expression as a result
    ??
     
  5. Jan 31, 2009 #4

    Mark44

    Staff: Mentor

    Where did you find this equation?

    After some thought, it doesn't seem to be true. I showed that when n = 1 it's true, namely that
    [tex]d/dx(x^0 e^{1/x}) = d/dx(e^{1/x}) = x^{-2}e^{1/x}[/tex]
    But when n = 2, you have
    [tex]d^2/dx^2(x e^{1/x}) = d/dx (x^{-2}e^{1/x}) [/tex]
    [tex]= e^{1/x}(1/x^4 + 2/x^3)[/tex]
    and this doesn't fit the pattern.

    Things will only get worse for higher derivatives, since you'll get an additional term for each additional derivative.
     
  6. Jan 31, 2009 #5

    marcus

    User Avatar
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    You have made a mistake taking the derivative. Incorrect application of the product rule or the quotient rule.
    Set the (-1)n aside and try again to take the derivative of [itex]\frac{e^{1/x}}{x^{n+1}}[/itex]

    This is the product of two pieces [itex]e^{1/x}[/itex] and [itex]\frac{1}{x^{n+1}}[/itex]

    The derivative of the first piece is [itex]-\frac{e^{1/x}}{x^{2}}[/itex]
    The derivative of the second piece is [itex]-(n+1)\frac{1}{x^{n+2}}[/itex]

    How about you apply the product rule and tell us what the derivative of gn is?

    If you can just get the derivative of e1/x/xn+1 right, then I think the rest will go OK.
    BTW the quotient rule is just a poorly disguised version of the product rule. You really only need one rule and the product rule will work in either case. Correctly applied they both give the same answer. You must learn at least to apply the product rule correctly
     
    Last edited: Jan 31, 2009
  7. Jan 31, 2009 #6
    is this correct??
    what to do next??
    [itex]
    g_n '(x) = ( - 1)^n \frac{{ - \frac{1}{{x^2 }}e^{1/x} x^{n + 1} - (n + 1)x^n e^{1/x} }}{{x^{2n + 2} }}
    [/itex]
     
  8. Jan 31, 2009 #7

    marcus

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    well, let's see. I will apply the product rule and see if what you have is right.

    (pq)' = p'q + p q'

    ((e1/x)( 1/xn+1))' = (-e1/x/x2) (1/xn+1) + (e1/x)(-(n+1)/xn+2)

    = -e1/x/xn+3 - (n+1) e1/x/xn+2

    = -(n + 1 + 1/x) e1/x/xn+2

    You can put back in the factor of (-1)n.

    Now is this what you got? I can't read your "itex" text, it is too small, please use plain "tex". I will copy your answer but with "tex":
    ==quote==
    is this correct??

    [tex]
    g_n '(x) = ( - 1)^n \frac{{ - \frac{1}{{x^2 }}e^{1/x} x^{n + 1} - (n + 1)x^n e^{1/x} }}{{x^{2n + 2} }}
    [/tex]
    ==endquote==

    Yes! It looks right! You just didn't simplify it, by canceling stuff.

    OK now we know that (gn(x))' = (n + 1 + 1/x) gn+1(x)

    Make sure you understand that much.
     
    Last edited: Jan 31, 2009
  9. Jan 31, 2009 #8

    marcus

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    OK now we know that (gn(x))' = (n + 1 + 1/x) gn+1(x).

    From here it should be a straight shot.
    For convenience let's write D for d/dx.
    Now just successively differentiate fn+1

    Dfn+1 = fn + xDfn
    D2fn+1 = 2Dfn + xD2fn
    D3fn+1 = 3D2fn + xD3fn
    D4fn+1 = 4D3fn + xD4fn
    Dn+1fn+1 = (n+1)Dnfn + xDn+1fn

    I think you have done that. Make sure you understand so far. Do it again, I would suggest, for good measure. It is now easy to finish solving. Try it. You don't have to depend, I think. Any questions about what's been done?
     
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