Mclaurin series for 10th derivative of arctan(x^2/6) at x=0

[thebenji]

How do i use mclaurin series to find high-numbered derivatives of functions?

for example:

the 10th derivative of arctan(x^2/6) at x=0

or

the 9th derivative of [cos(4x^2)-1]/x^3 at x=0

?

 

[tkhunny]

The first derivative might give more clues than the original function.

\(\displaystyle \frac{d}{dx}atan(\frac{x^{2}}{6})\;=\;\frac{12x}{x^{4}+36}\)

What does that series look like?

Perhaps you discussed a specific idea in class?

 

[thebenji]

I don't see anything.

I know that

arctan(x) = summation(1-)^n*[x^(2n+1)]/(2n+1).

Am I supposed to use that somehow?