Problem V$.$ Neural Network $(10$ points$)$

Consider a two$-$layer neural network described by following equations:

$a_1=w^{(1)}x,a_2=w^{(2)}z,z=h(a_1),y=(a_2)$

where $x$ and $y$ are the input and output, respectively, of the neural network, $h(*)$ is a

nonlinear function, and $(*)$ is the sigmod function.

$(1)$ Please derive the following gradients: $\frac{\text{d}\text{e}\text{l}\text{y}}{del{w}^{(1)}},\frac{\text{d}\text{e}\text{l}\text{y}}{del{w}^{(2)}},\frac{\text{d}\text{e}\text{l}\text{y}}{del{a}_1},\frac{\text{d}\text{e}\text{l}\text{y}}{del{a}_2},$ and $\frac{\text{d}\text{e}\text{l}\text{y}}{\text{d}\text{e}\text{l}\text{x}}.$

$(2)$ Please derive the updating rules for $w^{(1)}$ and $w^{(2)}$ given the classification errors between $y$

and $t,$ where $t$ is the ground truth of the output $y.$