The output $z$ to an artificial neural network with two inputs $(x,y)$ is given by

$z=(a_{0}u+b_{0}v+c_0)$

with $u=(a_{1}x+b_{1}y+c_1)$ and $v=(a_{2}x+b_{2}y+c_2),$ where $a_i,b_i,c_i$ are constant parameters and $(s)$ is a differentiable activation function.

$($a$)$ Draw a tree diagram on how $z$ is related to $x,y.$

$($b$)$ Derive a formula for $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{x}}$ and $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{y}}.$ Leave answer in terms of ${}^{\text{'}}$ with appropriate arguments.