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Math Help - 10th derivative

  1. #1
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    10th derivative

    Compute the 10th derivative of



    at x=0
    f^10(0)=?

    i think im supposed to use the Maclaurin series
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by dat1611 View Post
    Compute the 10th derivative of



    at x=0
    f^10(0)=?

    i think im supposed to use the Maclaurin series
    Use the McLaurin series for \arctan(u) and substitute u=x^2/4.

    CB
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    i dont know how im supposed to get a number answer out of it
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    Quote Originally Posted by dat1611 View Post
    i dont know how im supposed to get a number answer out of it
    You get the McLaurin series up to the term in x^{10} then you can differentiate term by term ten times (which is easy as the terms with exponent of x less than 10 will be eliminated by the time you have taken the 10-th derivative) and then set x=0 to evaluate (the terms with powers greater than 10 when x is set to zero will be zero so there was no need to look at them).

    CB
    Last edited by mr fantastic; August 21st 2009 at 03:38 AM. Reason: Fixed a latex exponent
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    Im not sure how to do the Maclaurin series
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    Quote Originally Posted by dat1611 View Post
    Im not sure how to do the Maclaurin series
    You should know the series:

    \arctan(u)=u-{{u^3}\over{3}}+{{u^5}\over{5}}-{{u^7}\over{7}}+ {{u^9}\over{9}}..

    you will not need the terms beyond u^5

    CB
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    how do i get a number if i plug in x^2/4
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    Quote Originally Posted by CaptainBlack View Post
    You should know the series:

    \arctan(u)=u-{{u^3}\over{3}}+{{u^5}\over{5}}-{{u^7}\over{7}}+ {{u^9}\over{9}}..

    you will not need the terms beyond u^5

    CB
    Quote Originally Posted by dat1611 View Post
    how do i get a number if i plug in x^2/4
    Substitute x^2/4 for u to get:

    \arctan(x^2/4)=\frac{x^2}{4}-{{x^6}\over{3.4^3}}+{{x^{10}}\over{5.4^5}}-{{x^{14}}\over{7.4^7}} + ..

    truncate after the term with x^{10} as you won't need them (Why is that?) and diffterentiate 10 times (you do that), then substitute 0 for x in what you have left (if anything).

    CB
    Last edited by CaptainBlack; August 21st 2009 at 01:21 PM. Reason: correct the typos
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  9. #9
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    so i differntiated x^2/4- x^6/12 + x^10/20 ten times and all i got is 181440 which is not right
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  10. #10
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     \arctan(u) = u - \frac{u^{3}}{3} + \frac{u^{5}}{5} - \frac{u^{7}}{7} + ...

     \arctan (x^{2}/4) = \frac{x^{2}}{4} - \frac{x^{6}}{3\cdot4^3} + \frac{x^{10}}{5 \cdot 4^{5}} + \frac{x^{14}}{7 \cdot4^{7}} + ...


     \frac{d^{n}}{dx^{n}} x^{k} = \frac{k!}{(k-n)!} x^{k-n}

    The 10th derivatives of the first two terms are zero for any value of x, and the 10th derivative of every term beyond x^{10} evaluates to zero at x=0.

    so  \frac{d^{10}}{dx^{10}} \arctan \Big(\frac{x^{2}}{4}\Big) = \frac{1}{5 \cdot 4^5} \frac{d^{10}}{dx^{10}} x^{10} = \frac{1}{5 \cdot 4^{5}}\frac{10!}{0!} x^{0} = \frac{2835}{4}

    which of course at x=0 is \ \frac{2835}{4}
    Last edited by Random Variable; August 21st 2009 at 09:53 AM.
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