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.

$\displaystyle f``(x)=\frac{f`(x)}{x}-4x^2f(x)$

Thanks.

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- Mar 23rd 2014, 11:06 AMkicma100th 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.

$\displaystyle f``(x)=\frac{f`(x)}{x}-4x^2f(x)$

Thanks. - Mar 23rd 2014, 04:40 PMromsekRe: 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 - Mar 24th 2014, 04:48 AMkicmaRe: 100th derivative in 0 of sin(x^2)
I don't really understand how did you (them) do that.

- Mar 24th 2014, 11:07 AMjohngRe: 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.

Attachment 30502 - Mar 24th 2014, 11:39 AMkicmaRe: 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.

- Mar 25th 2014, 08:58 AMhollywoodRe: 100th derivative in 0 of sin(x^2)
You should be able to prove romsek's formula by mathematical induction.

- Hollywood - Apr 18th 2014, 06:51 AMkicmaRe: 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