Compute the 9th derivative of f(x) = arctan(x^3/3) at x=0. please i need an answer?

Question

Compute the 9th derivative of f(x) = arctan(x^3/3) at x=0. please i need an answer?

Answer 1

What's the first derivative of arctan x when x =0.

Looks like a trick question x to the 3/3 well 3/3 =1 so it's arctan(x^1).

also the when x=0 seems to be a trick.

Sounds like quite an easy question.

Answer 2

The first derivative is

[ 1 / (1 + (x^3 / 3)^2) ] * x^2 =

9x^2 / (1 + x^6)

Second derivative is

[18x / (1 + x^6)]  - 9x^2(1+x^6)^-2 (6x^5) =

[18x / (1 + x^6)]  - 54x^7(1+x^6)^-2  =

[9x^2 / (1 + x^6)] [ 2/x - 6x^5/(1+x^6) ]  =

f ' (x) * [ 2/x - 6x^5/(1+x^6) ]

After a few more you might notice a pattern.  If you can show that all of the derivatives are a multiple or linear combination of the previous derivatives, and that f ' (0) = 0, then the 9th derivative should be the same.

Answer 3

If you need someone to actually show the work you may be waiting a while. According to MATLAB, the ninth derivative at x=0 is exactly

-4480