Results 1 to 7 of 7

Math Help - 100th derivative in 0 of sin(x^2)

  1. #1
    Newbie
    Joined
    Jan 2013
    From
    Croatia
    Posts
    24

    100th derivative in 0 of sin(x^2)

    I need to find 100th derivative of sin(x^2) in 0.

    I came up with this, but don't know what to do next.
     f``(x)=\frac{f`(x)}{x}-4x^2f(x)

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,572
    Thanks
    1009

    Re: 100th derivative in 0 of sin(x^2)

    $\dfrac{d^{(n)}}{{dx}^{(n)}} \sin(x^2) |_{x=0}= \left \{
    \begin{array}{ll}
    (-1)^n \dfrac{(4n+2)!}{(2n+1)!}~:n=(4k+2), k=0,1,2, \dots 24 \\
    0~:~ \text{else}
    \end{array}\right.$

    https://oeis.org/A024343
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jan 2013
    From
    Croatia
    Posts
    24

    Re: 100th derivative in 0 of sin(x^2)

    I don't really understand how did you (them) do that.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Dec 2012
    From
    Athens, OH, USA
    Posts
    643
    Thanks
    258

    Re: 100th derivative in 0 of sin(x^2)

    Hi,
    If you know about power series, this is not too hard. Otherwise, I can't see an easy way to do it. The attachment shows the power series argument in some detail.

    100th derivative in 0 of sin(x^2)-mhfcalc41.png
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jan 2013
    From
    Croatia
    Posts
    24

    Re: 100th derivative in 0 of sin(x^2)

    Hmm, there should be a way without power series because we haven't done that yet. Leibniz rule should probably be used.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    Mar 2010
    Posts
    993
    Thanks
    244

    Re: 100th derivative in 0 of sin(x^2)

    You should be able to prove romsek's formula by mathematical induction.

    - Hollywood
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Jan 2013
    From
    Croatia
    Posts
    24

    Re: 100th derivative in 0 of sin(x^2)

    Ok, I solved it. Trick is to write it as xy''=y'-4yx^3 and then use Leibniz rule for (n-1)th derivative
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. difference between intrinsic derivative and covariant derivative
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 16th 2013, 01:58 AM
  2. Replies: 4
    Last Post: October 17th 2012, 06:51 AM
  3. contuous weak derivative $\Rightarrow$ classic derivative ?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 22nd 2011, 02:37 AM
  4. Replies: 0
    Last Post: January 24th 2011, 11:40 AM
  5. Replies: 2
    Last Post: November 6th 2009, 02:51 PM

Search Tags


/mathhelpforum @mathhelpforum