Question

Compute the 8th derivative of ...

Answer 1

\[\large f(x)=\frac{\cos(3x^3)-1}{x^{4}}\]

Answer 2

at x = 0

Answer 3

lol, this will take a long time

Answer 4

There is a hint to use MacLaurin series for f(x) which i understand would help but i can't see any pattern coming up on the derivatives of f(x)

Answer 5

even so, that takes a long time

Answer 6

(i've only gone up to the 2nd derivative tho)

Answer 7

expand cos(x^3) this way ... subtract 1 and divide by x^4

the coefficient of third term is your 8th derivative.

Answer 8

http://www.wolframalpha.com/input/?i=expand+cos%28x%29

Answer 9

so you're allowed to take the series of cos(3x^3) and at each term you -1 and /(x^4) ?

Answer 10

no not each term. just remove 1 from the expansion of cos(x^3)
http://www.wolframalpha.com/input/?i=expand+cos%28x^3%29-1+at+x%3D0

Answer 11

The expansion is as
\[ -\frac{x^2}{2}+\frac{x^8}{24}-\frac{x^{14}}{720}+\frac{x^{20}}{40320}-\frac{x^{26}}{3628800}+\frac{x^{32}}{479001600}-\frac{x^{38}}{87178291200}+O[x]^{41} \]
the 8th derivative is 1/24 at x=0

Answer 12

gotta go out for a while.

Answer 13

i think i got it :) thanks :)

Answer 14

hold on ... the 8th derivative is
\[ {8! \over 24} = 1680\]

Answer 15

the derivative should be
\[ 1680-168168 x^6+125970 x^{12}-\frac{312455 x^{18}}{18}+\frac{58435 x^{24}}{66}-\frac{370481 x^{30}}{16380}+O[x]^{33} \]

Answer 16

this whole should be equal to
\[ \frac{6652800 \left(1+\text{Cos}\left[x^3\right]\right)}{x^{12}}+\frac{14515200 \text{Sin}\left[x^3\right]}{x^9}+\\
\frac{1693440 \left(-9 x^4 \text{Cos}\left[x^3\right]-6 x \text{Sin}\left[x^3\right]\right)}{x^{10}}+\\
\frac{58800 \left(-180 x^2 \text{Cos}\left[x^3\right]+81 x^8 \text{Cos}\left[x^3\right]+324 x^5 \text{Sin}\left[x^3\right]\right)}{x^8}-\\
\frac{376320 \left(-54 x^3 \text{Cos}\left[x^3\right]-6 \text{Sin}\left[x^3\right]+27 x^6 \text{Sin}\left[x^3\right]\right)}{x^9}+\\
\frac{560 \left(-360 \text{Cos}\left[x^3\right]+17820 x^6 \text{Cos}\left[x^3\right]-729 x^{12} \text{Cos}\left[x^3\right]+9720 x^3\\
\text{Sin}\left[x^3\right]-7290 x^9 \text{Sin}\left[x^3\right]\right)}{x^6}-\frac{6720 \left(-360 x \text{Cos}\left[x^3\right]+1620 x^7 \\
\text{Cos}\left[x^3\right]+2160 x^4 \text{Sin}\left[x^3\right]-243 x^{10} \text{Sin}\left[x^3\right]\right)}{x^7}+\frac{1}{x^4}\left(771120 x^4 \\
\text{Cos}\left[x^3\right]-694008 x^{10} \text{Cos}\left[x^3\right]+6561 x^{16} \text{Cos}\left[x^3\right]+60480 x \text{Sin}\left[x^3\right]-1360800 x^7 \\
\text{Sin}\left[x^3\right]+122472 x^{13} \text{Sin}\left[x^3\right]\right)-\frac{1}{x^5}32 \left(136080 x^5 \text{Cos}\left[x^3\right]-30618 x^{11} \\
\text{Cos}\left[x^3\right]+30240 x^2 \text{Sin}\left[x^3\right]-119070 x^8 \text{Sin}\left[x^3\right]+2187 x^{14} \text{Sin}\left[x^3\right]\right) \]