## Some derivatives

In the book it said, let
$\displaystyle \displaystyle f(x)=\frac{x^{p-1}}{(p-1)!}(1-x)^p(2-x)^p\cdots (n-x)^p$
where $\displaystyle p$ is a prime number and $\displaystyle p>n$. Then it said that when $\displaystyle i\ge p$, the coefficients of $\displaystyle f^{(i)} (x)$ are multiples of $\displaystyle p$. I can't see why it is true. Can someone elaborate? Thanks.