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

• Mar 23rd 2014, 12:06 PM
kicma
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.
• Mar 23rd 2014, 05:40 PM
romsek
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
• Mar 24th 2014, 05:48 AM
kicma
Re: 100th derivative in 0 of sin(x^2)
I don't really understand how did you (them) do that.
• Mar 24th 2014, 12:07 PM
johng
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.

Attachment 30502
• Mar 24th 2014, 12:39 PM
kicma
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.
• Mar 25th 2014, 09:58 AM
hollywood
Re: 100th derivative in 0 of sin(x^2)
You should be able to prove romsek's formula by mathematical induction.

- Hollywood
• Apr 18th 2014, 07:51 AM
kicma
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