Question $3.$ Consider a neural network model with one hidden layer and the following

properties:

The size of the input is $2.$

The size of hidden layer is $3$ with each node has activation function Sigmoid:

$S(x)=\frac{1}{1+e^{-x}}.$

The output has size $1$ with activation function $tanh(x).$

$($a$)$ Draw a computational graph for this neural network.

$($Please write the answers on a piece of paper so that I can see the true graph and derivations in part $($b$).$ Thank you in advance!$)$

$($b$)$ The binary cross$-$entropy loss function for a label vector $y$ and the prediction vector

hat$(y)$ is defined as

$L(y,$hat$(y))=-\frac{1}{l}{\displaystyle \underset{i=1}{\overset{l}{}}}[y_i*lo{g}_2(hat(y{)}_i)+(1-y_i)*lo{g}_2(1-$hat$(y{)}_i)]$

Note that hat$(y{)}_i$ is the output of the neural network for input $x_i.$

Calculate the gradient of $L$ with respect to the weights of the first layer $W^{(1)}.$ You

can use a matrix notation.