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

1. 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.

2. 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

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

I don't really understand how did you (them) do that.

4. 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.

5. 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.

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

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

- Hollywood

7. 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