- #1

kamil

- 6

- 0

## Homework Statement

Compute the 9th derivative of

[tex]f(x) = \frac{\cos\left(3 x^{4} \right) - 1}{x^{7}}[/tex]

at x=(0)

## Homework Equations

[tex]f(x)=\sum^{\infty}_{n=0} \frac{f^{(n)}(c)}{n!}x^n[/tex]

[tex]cos(x)=\sum^{\infty}_{n=0} \frac{(-1)^{n}}{(2n)!}x^{2n}[/tex]

## The Attempt at a Solution

The correct answer is 1224720.

I've already read this thread(https://www.physicsforums.com/showthread.php?t=362369&highlight=Maclaurin), and I think I know how to do these kind of problems. However, I have a problem with that one since it has [tex]{-1}}[/tex]

What I did is write the power series for cos(x),

[tex]cos(x)=\sum^{\infty}_{n=0} \frac{(-1)^{n}}{(2n)!}x^{2n}[/tex]

Substitute [tex] 3 x^{4}[/tex] , substract 1, and divide by [tex] x^{7}[/tex].

And getting:

[tex]\frac{\cos\left(3 x^{4} \right) - 1}{x^{7}}=\sum^{\infty}_{n=0} \frac{\frac{(-1)^{n}3^{2n}}{(2n)!}x^{8n}-1}{x^7}[/tex]

But after that I don't know.