$ \frac{d}{dw} = \frac{1}{N} \sum_{n=1}^N (y_n - x_n^Tw)x_n + 2\lambda w $
$ = \frac{1}{N} \sum_{n=1}^N y_nx_n - x_n^Twx_n + 2 \lambda w$
can i shift w anyhow I like ? eg
$?= \frac{1}{N} \sum_{n=1}^N y_nx_n - x_nx_n^Tw + 2 \lambda w $
Is this correct ? Why yes or why no ?
If $x_n^Tw$ is a scalar, then we have
$$(x_n^Tw)x_n = x_n(x_n^Tw)$$
Additional remark:
$\frac{d}{dw}$ is not relevant.
try to use paranthesis reasonably generously.
$$\sum_{n=1}^N (y_n - x_n^Tw)x_n = \sum_{n=1}^N (y_nx_n - (x_n^Tw)x_n)$$