You can start from the 'explicit definition' of the 'Bernoulli polynomial of index n'...
$\displaystyle B_{n}(x)= \sum_{k=0}^{n} \binom{n}{k}\ B_{k}\ x^{n-k}$ (1)
... where $\displaystyle B_{k}$ isw the 'Bernoulli number of index k'. Deriving (1) You obtain...
$\displaystyle B_{n}^{'}(x)= \sum_{k=0}^{n} \binom{n}{k}\ B_{k}\ (n-k)\ x^{n-k-1}= $
$\displaystyle = n\ \sum_{k=0}^{n-1} \binom{n-1}{k}\ B_{k}\ x^{n-k-1} = n\ B_{n-1}(x)$ (2)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$