Linear model of the modified error function
Yes, you can do that.

$$E(w)=1/2\sum_{n=1}^{N}\{y(x_n,w)-t_n\}^2+\dfrac \lambda2||w||^2$$

$$\begin{align} \frac{\partial E(w)}{\partial w} &amp;= \frac{\partial \left( 1/2\sum_{n=1}^{N}\{y(x_n,w)-t_n\}^2+\dfrac \lambda2||w||^2 \right)}{\partial w} \\ &amp;= \frac{\partial \left( 1/2(y({\bf x},{\bf w} )-{\bf t})^T(y({\bf x},{\bf w} )-{\bf t})+\dfrac \lambda2{\bf w}^T{\bf w} \right)}{\partial {\bf w}} \\ &amp;= \frac{\partial \left( 1/2(y({\bf x},{\bf w} )-{\bf t})^T(y({\bf x},{\bf w} )-{\bf t})\right)}{\partial {\bf w}}+\frac{\partial \left( \dfrac \lambda2{\bf w}^T{\bf w} \right)}{\partial {\bf w}} \\ &amp;= \left( \frac{\partial y({\bf x},{\bf w} )}{\partial {\bf w} } \right)^T \left( y({\bf x},{\bf w} )-{\bf t} \right) + \lambda{ \bf w }\end{align}$$