Question

if y=cos(2x), determine d^80y/dx^80

Answer 1

It's cyclical-ish.

1) (d/dx) cos(2x) = -2sin(2x)
2) (d/dx) -2sin(2x) = -4cos(2x)
3) (d/dx) -4cos(2x) = 8sin(2x)
4) (d/dx) 8sin(2x) = 16cos(2x)

No the thinking about it. \[2^{4}cos(2x)\]
After 8 \[2^{8}cos(2x)\]
After 16 \[2^{16}cos(2x)\]
After 32 \[2^{32}cos(2x)\]

Do you see it?

Answer 2

Yes, so it's \[2^{80}\sin(2x)\]
Thank you~ :D

Answer 3

Why did you switch it to sine? Any multiple of 4 will be a positive cosine.

Answer 4

ohh... when would it switch to sine?

Answer 5

That's why I wrote out the first 4.

1 more than a multiple of 4 leads to negative sine
2 more than a multiple of 4 leads to negative cosine
3 more than a multiple of 4 leads to positive sine
Multiple of 4 leads to positive cosine

Answer 6

Thank you again, I'll write it down for future reference. If it goes onto 5, would that be back to negative sine?

Answer 7

Again, that's why I wrote it all that way. I did not stick to a specific case. I wrote it for ALL possibilities. Read it more carefully and you will see it.