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

Aug 21, 2014

The answer is ${y}^{\left(108\right)} = \cos \left(x\right)$

$\sin x$ and $\cos x$ has a 4 derivative cycle:

$\frac{d}{\mathrm{dx}} \cos x = - \sin x$
$\frac{d}{\mathrm{dx}} - \sin x = - \cos x$
$\frac{d}{\mathrm{dx}} - \cos x = \sin x$
$\frac{d}{\mathrm{dx}} \sin x = \cos x$

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.

$\frac{d}{\mathrm{dx}} \cos x = - \sin x$ (mod 4=1)
$\frac{d}{\mathrm{dx}} - \sin x = - \cos x$ (mod 4=2)
$\frac{d}{\mathrm{dx}} - \cos x = \sin x$ (mod 4=3)
$\frac{d}{\mathrm{dx}} \sin x = \cos x$ (mod 4=0)

So, our answer is $\cos x$.