Question

If y=cos x, what is y^6 (x)

Answer 1

You mean \(\large\rm y^{(6)}(x)\) yes? As in 6th derivative?

Answer 2

Yes

Answer 3

Take some derivatives, look for a pattern :D

Answer 4

Err I should say this...
when they give you a similar type problem and ask you to find \(\large\rm y^{(108)}(x)\), then it's imperative that you recognize a pattern,
because you don't want to take 108 derivatives.

Answer 5

But in this case, taking 6 derivatives in a row should be pretty easy.

Answer 6

Do you remember your sine and cosine derivatives?

Answer 7

Well I know that sinx=cos x and cosx=-sinx. Is this what what mean?

Answer 8

Yes.\[\large\rm \sin x\quad\to\quad \cos x\quad\to\quad-\sin x\quad\to\quad-\cos x\quad\to\quad \sin x\]

Answer 9

Is it the same as f(x)=x^(2) which equals 2x?

Answer 10

I'm not sure what you're asking..

Answer 11

Sorry Im looking at something else.

Answer 12

Anyway, notice that the trig derivatives come full circle after taking 4 of them.
See how I started with sine and ended with sine after 4 derivatives?

Answer 13

Yes

Answer 14

So if you take 4 derivatives of cosine, you'll get back to cosine.
Which means we really only need to take 2 derivatives.
6 derivatives of cosine is the same as 2 derivatives of cosine.

Answer 15

Oh okay so how would that look in terms of the problem?

Answer 16

\[\large\rm y=\cos x\]\[\large\rm y'=-\sin x\]\[\large\rm y''=-\cos x\]\[\large\rm y'''=\sin x\]\[\large\rm y^{(4)}=\cos x\]So four derivatives got us back to cosine.
Take two more derivatives.

Answer 17

so y^(5)=-sin x
y^(6)=-cox x ?

Answer 18

Yay good job

Answer 19

Wow really! Thank you for explaining that for me.

Answer 20

np