  Let $k[X]=k[x_0,...,x_{n-1}]$ and $k[Y]=k[y_0,...,y_{n-1}]$ be the polynomial
rings in $n\geqslant 3$ variables over a field $k$ of characteristic zero
containing the $n$-th roots of unity. Let $d$ be the cyclotomic derivation of
$k[X]$, and let $\Delta$ be the factorisable derivation of $k[Y]$ associated
with $d$, that is, $d(x_j)=x_{j+1}$ and $\Delta(y_j)=y_j(y_{j+1}-y_j)$ for all
$j\in\mathbb Z_n$. We describe polynomial constants and rational constants of
these derivations. We prove, among others, that the field of constants of $d$
is a field of rational functions over $k$ in $n-\f(n)$ variables, and that the
ring of constants of $d$ is a polynomial ring if and only if $n$ is a power of
a prime. Moreover, we show that the ring of constants of $\Delta$ is always
equal to $k[v]$, where $v$ is the product $y_0... y_{n-1}$, and we describe the
field of constants of $\Delta$ in two cases: when $n$ is power of a prime, and
when $n=p q$.
