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Math Help - Compute the 10th derivative of ...

  1. #1
    s3a
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    Compute the 10th derivative of ...

    The question is attached. I know the power series representation of cos(x) and therefore can get cos(6x^2)/x^2 but I don't know what to do about the -1.

    Any help would be greatly appreciated!
    Thanks in advance!
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  2. #2
    Super Member Random Variable's Avatar
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    Start with the known Maclaurin series of  \cos x

     \cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + ...

    then  \cos 6x^{2} = 1 - \frac{(6x^{2})^{2}}{2!} + \frac{(6x^{2})^{4}}{4!} - \frac{(6x^{2})^{6}}{6!} + ... = 1  - \frac{6^{2}x^{4}}{2!} + \frac{6^{4}x^{8}}{4!} - \frac{6^{6}x^{12}}{6!} + ...

    and \cos 6x^{2} -1  = -\frac{6^{2}x^{4}}{2!} + \frac{6^{4}x^{8}}{4!} -  \frac{6^{6}x^{12}}{6!} + ...

    finally  \frac{\cos 6x^{2}-1}{x^{2}} = -\frac{6^{2}x^{2}}{2!} + \frac{6^{4}x^{6}}{4!} -  \frac{6^{6}x^{10}}{6!} +  ...


    For a Macluarin series, the coefficent of the  x^{10} term is  \frac{f^{(10)}(0)}{10!}

    The coefficient of the  x^{10} term in the above Macluarin series is  -\frac{6^{6}}{6!}

    so  \frac{f^{(10)}(0)}{10!} =  -\frac{6^{6}}{6!}

    EDIT: and  f^{(10)}(0) = - \frac{6^{6} 10!}{6!} =  - 235146240
    Last edited by Random Variable; May 17th 2010 at 02:01 PM.
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  3. #3
    s3a
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    How do you get from the before-last step to the last one? (From the one that says "so" to the one with the emoticon)
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  4. #4
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    the 10! came from the exponent on  \frac{6^{6}x^{10}}{6!} . If you were to take the derivative of that 10 times, you would get  \frac{6^{6} 10!}{6!}
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  5. #5
    Super Member Random Variable's Avatar
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    Quote Originally Posted by s3a View Post
    How do you get from the before-last step to the last one? (From the one that says "so" to the one with the emoticon)
    It should be  f^{(10)} (0) = - \frac{6^{6} 10!}{6!}
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  6. #6
    s3a
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    Oh that 10! was also tricky but I get it now! Thanks!
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