$[30$ marks$]$ Linear regression aims to fit a linear function $h_{}(vec(x))=$vec$({)}^T*vec(x)={}_0+{}_1{x}_1+{}_2{x}_2+$dotsdots$+{}_n{x}_n$ based on the training set $D=\{\begin{array}{ccc}\text{v}\text{e}\text{c}& (x{)}^{(i)},& y^{(i)}\\ i=\end{array}1,2,$dots,$m$ to predict the numerical output $y$ for a new input vector vec$(x).$ Please derive the update rules for both stochastic and batch gradient descent methods to iteratively update vec$(),$ aiming to minimize the least squares cost function $J(vec())$ of linear regression.

$J(vec())=\frac{1}{2}{\displaystyle \underset{i=1}{\overset{m}{}}}[h_{}(vec(x{)}^{(i)})-y^{(i)}{]}^2$