# Your task is to replace $???$

import numpy as np

# First we set the state of the network

$\backslash$sigma $=$ np$.$tanh

w$1=1\mathrm{.}3$

b$1=-0\mathrm{.}1$

# Then we define the neuron activation.

def a$1($a$0)$:

z $=$ w$1*$ a$0+$ b$1$

return $\backslash$sigma $($z$)$

# Experiment with different values of x below.

x $=1$

print$(($a$1($x$)-0)**2)$

# First define our sigma function.

sigma $=$ np$.$tanh

# Next define the feed$-$forward equation.

def a$1($w$1,$ b$1,$ a$0)$ :

z $=$ w$1*$ a$0+$ b$1$

return sigma$($z$)$

# The individual cost function is the square of the difference between

# the network output and the training data output.

def C $($w$1,$ b$1,$ x$,$ y$)$ :

return $($a$1($w$1,$ b$1,$ x$)-$ y$)**2$

# This function returns the derivative of the cost function with

# respect to the weight.

def dCdw $($w$1,$ b$1,$ x$,$ y$)$ :

z $=???$

dCda $=???$ # Derivative of cost with activation

dadz $=1/$np$.$cosh$($z$)**2$ # derivative of activation with weighted sum z

dzdw $=???$ # derivative of weighted sum z with weight

return $???$ # Return the chain rule product.

# This function returns the derivative of the cost function with

# respect to the bias.

# It is very similar to the previous function.

# You should complete this function.

def dCdb $($w$1,$ b$1,$ x$,$ y$)$ :

z $=???$

dCda $=???$

dadz $=???$

# Change the next line to give the derivative of

# the weighted sum, z$,$ with respect to the bias, b$.$

dzdb $=???$

return $???$

# Define the activation function.

sigma $=$ np$.$tanh

# Let's use a random initial weight and bias.

W $=$ np$.$array$([[-0\mathrm{.}94529712,-0\mathrm{.}2667356,-0\mathrm{.}91219181],$

$[2\mathrm{.}05529992,1\mathrm{.}21797092,0\mathrm{.}22914497]])$

b $=$ np$.$array$([0\mathrm{.}61273249,1\mathrm{.}6422662])$

# define our feed forward function

def a$1($a$0)$ :

# Notice the next line is almost the same as previously,

# except we are using matrix multiplication rather than scalar multiplication

z $=???$

# Everything else is the same though,

return sigma$($z$)$

# Next, if a training example is$,$

x $=$ np$.$array$([0\mathrm{.}7,0\mathrm{.}6,0\mathrm{.}2])$

y $=$ np$.$array$([0\mathrm{.}9,0\mathrm{.}6])$

# Then the cost function is$,$

d $=???$ # Vector difference between observed and expected activation

C $=???$ # Absolute value squared of the difference.

# First define our sigma function.

sigma $=$ np$.$tanh

# Next define the feed$-$forward equation.

def a$1($w$1,$ b$1,$ a$0)$ :

z $=???$

return sigma$($z$)$

# This function returns the derivative of the cost function with

# respect to the weight.

def dCdw $($w$1,$ b$1,$ x$,$ y$)$ :

dCda $=???$ # Derivative of cost with activation

dadz $=???$ # derivative of activation with weighted sum z

J $=???$

dzdw $=???$# derivative of weighted sum z with weight

J $=???$

return J # Return the chain rule product.

# This function returns the derivative of the cost function with

# respect to the bias.

# It is very similar to the previous function.

# You should complete this function.

def dCdb $($w$1,$ b$1,$ x$,$ y$)$ :

dCda $=???$

dadz $=???$

# Change the next line to give the derivative of

# the weighted sum, z$,$ with respect to the bias, b$.$

dzdb $=???$

return dCda $*$ dadz $*$ dzdb