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Math Help - Mathematical induction.

  1. #1
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    Exclamation Mathematical induction.

    How can we show this by using mathematical induction

    (d^n\dx^n )x^n = n!
    Last edited by mr fantastic; February 23rd 2011 at 04:23 AM. Reason: Title.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by mariama View Post
    How can we show this by using mathematical induction

    (d^n\dx^n )x^n = n!

    First step . We have

    \dfrac{d^1}{dx^1}x^1=1=1!

    so, the equality is true for n=1

    What difficulties have you had in the second step?

    Remark . If the question is for n\geq 0 then, take into account that:

    \dfrac{d^0}{dx^0}f(x)=f(x) and 0!=1


    Fernando Revilla
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  3. #3
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    so , Should I plug n+1 instead of n in the next step ?? or how can i make it
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by mariama View Post
    so , Should I plug n+1 instead of n in the next step ?? or how can i make it
    Suppose:

    \dfrac{d}{dx^n}x^n=n!

    Then, prove:

    \dfrac{d}{dx^{n+1}}x^{n+1}=(n+1)!


    Fernando Revilla
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  5. #5
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    deleted
    Last edited by Archie Meade; February 23rd 2011 at 04:41 AM.
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  6. #6
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    Quote Originally Posted by FernandoRevilla View Post
    Suppose:

    \dfrac{d}{dx^n}x^n=n!

    Then, prove:

    \dfrac{d}{dx^{n+1}}x^{n+1}=(n+1)!


    Fernando Revilla

    Sorry , but i could not prove this
    \dfrac{d}{dx^{n+1}}x^{n+1}=(n+1)!
    Do you have a hint? or how can i start ?
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  7. #7
    Behold, the power of SARDINES!
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    Quote Originally Posted by mariama View Post
    Sorry , but i could not prove this
    \dfrac{d}{dx^{n+1}}x^{n+1}=(n+1)!
    Do you have a hint? or how can i start ?
    FernandoRevilla has outlined it for you. You need to show that n \implies n+1

    So start with
    \displaystyle \frac{d^{n+1}}{dx^{n+1}}x^{n+1}=\frac{d^n}{dx^n}\l  eft( \frac{d}{dx}x^{n+1}\right)
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  8. #8
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    Quote Originally Posted by TheEmptySet View Post
    FernandoRevilla has outlined it for you. You need to show that n \implies n+1

    So start with
    \displaystyle \frac{d^{n+1}}{dx^{n+1}}x^{n+1}=\frac{d^n}{dx^n}\l  eft( \frac{d}{dx}x^{n+1}\right)
    So , it will be
    d^n\(dx^n). n+1(x^n)
    hence (d\dx^n).x^n= n!

    = n! (n+1) = (n+1)!

    Is this right ?
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