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

##### 1 Answer

Aug 21, 2014

The answer is

#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