Question

How do you find the 108th derivative of `y=cos(x)` ?

Answer 1

The answer is `y^((108))=cos(x)`

`sin x` and `cos x` has a 4 derivative cycle:

`d/(dx)cos x=-sin x`
`d/(dx)-sin x=-cos x`
`d/(dx)-cos x=sin x`
`d/(dx)sin x=cos x`

Rather than doing 108 derivatives, we need to calculate 108 modulus 4; this equals 0. Although remainder works for positive dividends, it's best to get used to modulus because this works for negative dividends. Modulus 4 will return either 0, 1, 2, or 3.

`d/(dx)cos x=-sin x` (mod 4=1)
`d/(dx)-sin x=-cos x` (mod 4=2)
`d/(dx)-cos x=sin x` (mod 4=3)
`d/(dx)sin x=cos x` (mod 4=0)

So, our answer is `cos x`.