I need help implementing the test$\_$derivative, to$\_$fraction, and is$\_$in$\_$fraction$\_$form functions$($in python$).$ They should pass the given tests. Code: import math

import random

class IllegalOperator$($Exception$)$:

pass

def calc$($op$,$left,right$)$:

if op$=="+"$:

return left$+$right

elif op$=="-"$:

return left$-$right

elif op$=="*"$:

return left$*$right

elif op $=="/"$:

return left$/$right

else:

raise IllegalOperator$($op$)$

def compute$($e$)$:

if isinstance$($e$,$tuple$)$:

# We have an expression.

op$,$l$,$r$=$e

# We compute the subexpressions.

ll$=$compute$($l$)$

rr$=$compute$($r$)$

# And on the basis of those,the whole expression.

return calc$($op$,$ll$,$rr$)$

else:

# base expression; just return the number.

return e

from numbers import Number

def isnumber$($e$)$:

return isinstance$($e$,$Number$)$

def isvariable$($e$)$:

return isinstance$($e$,$str$)$

def iscomposite$($e$)$:

return isinstance$($e$,$tuple$)$

def simplify$($e$)$:

if isinstance$($e$,$tuple$)$:

op$,$ l$,$ r $=$ e

ll$=$simplify$($l$)$

rr$=$simplify$($r$)$

if isnumber$($ll$)$ and isnumber$($rr$)$:

return calc$($op$,$ll$,$rr$)$

else:

return $($op$,$ll$,$rr$)$

else:

return e

varval $=\{\text{'}$x$\text{'}$: $3,\text{'}$y$\text{'}$: $8\}$

#@title Evaluating an expression with respect to a variable evaluation

def compute$($e$,$ varval$=\{\})$:

if isinstance$($e$,$tuple$)$:

op$,$left,right $=$ e

new$\_$e $=$ op$,$compute$($left$,$varval$),$compute$($right$,$varval$)$

return simplify$($new$\_$e$)$

elif e in varval:

return varval$[$e$]$

else:

return e

#@title Exercise: define $`$variables$`$

def variables$($e$)$:

var$\_$set $=$ set$()$

operation $=[\text{'}+\text{'},\text{'}-\text{'},\text{'}*\text{'},\text{'}/\text{'}]$

if iscomposite$($e$)$:

for i in e:

if iscomposite$($i$)$:

a $=$ variables$($i$)$

var$\_$set $=$ a $|$ var$\_$set

if isvariable$($i$)$ and i not in operation:

var$\_$set.add$($i$)$

return var$\_$set

else:

return $\{$e$\}$

#@title Exercise: implementation of value equality

def value$\_$equality$($e$,$f$,$num$\_$samples$=1000,$tolerance$=1$e$-6)$:

$"""$Return True if the two expressions self and other are numerically

equivalent. Equivalence is tested by generating

num$\_$samples assignments,and checking that equality holds

for all of them. Equality is checked up to tolerance,that is$,$

the values of the two expressions have to be closer than tolerance."""

vars $=$ variables$($e$)|$ variables$($f$)$

for i in range$($num$\_$samples$)$:

x $=\{$var: random.gauss$(0,10)$ for var in vars$\}$

if abs$($compute$($e$,$x$)-$compute$($f$,$x$))>$tolerance:

return False

return True

#@title Derivation of a leaf expression

def derivate$\_$leaf$($e$,$x$)$:

$"""$This function takes as input an expression e and a variable x$,$

and returns the symbolic derivative of e wrt$.$ x$,$as an expression."""

if isvariable$($x$)$ and isvariable$($e$)$:

if x in e:

return $1$

return $0$

#@title Implement $`$derivate$`$

def derivate$($e$,$x$)$:

$"""$Returns the derivative of e wrt x$."""$

if not iscomposite$($e$)$:

return derivate$\_$leaf$($e$,$x$)$

op$,$f$,$g $=$ e

df $=$ derivate$($f$,$x$)$ if iscomposite$($f$)$ else derivate$\_$leaf$($f$,$x$)$

dg $=$ derivate$($g$,$x$)$ if iscomposite$($g$)$ else derivate$\_$leaf$($g$,$x$)$

if op in $\text{'}+-\text{'}$:

return $($op$,$df$,$dg$)$

elif op $==\text{'}*\text{'}$:

return $(\text{'}+\text{'},(\text{'}*\text{'},$df$,$g$),(\text{'}*\text{'},$f$,$dg$))$

elif op $==\text{'}/\text{'}$:

return $(\text{'}/\text{'},(\text{'}-\text{'},(\text{'}*\text{'},$df$,$g$),(\text{'}*\text{'},$f$,$dg$)),(\text{'}*\text{'},$g$,$g$))$

def derivate$\_$approx$($f$,$x$,$varval,delta$=0\mathrm{.}0001)$:

$"""$Computes the derivative of f with respect to x$,$for a given delta,

using the $($f$($x $+$ delta$)-$ f$($x$))/$ delta method. $"""$

# This is f$($x$)$

f$\_$x $=$ compute$($f$,$varval$=$varval$)$

varval$\_$delta $=$ dict$($varval$)$

varval$\_$delta$[$x$]+=$ delta

# This is f$($x $+$ delta$)$

f$\_$x$\_$plus$\_$delta $=$ compute$($f$,$varval$=$varval$\_$delta$)$

return $($f$\_$x$\_$plus$\_$delta $-$ f$\_$x$)/$ delta

def similar$($x$,$y$,$epsilon$)$:

if x $<mtext\; id="EditorElement\_2855237"></mtext><mn\; id="EditorElement\_2855238">0</mn>$ and y $<mtext\; id="EditorElement\_2855247"></mtext><mn\; id="EditorElement\_2855248">0</mn>$:

return similar$(-$x$,-$y$,$epsilon$)$

if abs$($x $-$ y$)$ epsilon:

return True

else:

return max$($x$,$y$)/($min$($x$,$y$)+$ epsilon epsilon

#@title Implementation of $`$test$\_$derivative$`$

def test$\_$derivative$($f$,$df$,$x$,$delta$=0\mathrm{.}0001,$tolerance$=0\mathrm{.}01,$num$\_$tests$=1000)$:

$\text{'}\text{'}\text{'}$df$\_$found $=$ derivate$($f$,$x$)$

return value$\_$equality$($df$\_$found,df$)\text{'}\text{'}\text{'}$

s $=$ True

for i in range$($num$\_$tests$)$:

test$\_$point $=$ x $+$ delta $*(2*$ random.random$()-1)$

y $=($f$($test$\_$point $+$ delta$)-$ f$($test$\_$point$))/$ delta

if abs$($df$($test$\_$point$)-$ y$)>$ tolerance:

s $=$ False

break

return s

# Tests $10$ points: $`$test$\_$derivative$`$

f $=("+",("*",$"cat","cat"$),("*",$"dog","cat"$))$

df$1=("+",("*",2,$"cat"$),$"dog"$)$

df$2=("+",("*",2,$"cat"$),("*",$"dog","cat"$))$

assert test$\_$derivative$($f$,$df$1,$"cat"$)==$ True

assert test$\_$derivative$($f$,$df$2,$"cat"$)==$ False

assert test$\_$derivative$($f$,$df$1,$"dog"$)==$ False

assert test$\_$derivative$($f$,$df$1,$"donkey"$)==$ False

assert test$\_$derivative$($f$,0,$"donkey"$)==$ True

#@title Putting expressions in fraction form.

def to$\_$fraction$($e$)$:

$"""$Returns the expression e converted to fraction form."""

### SOLUTION HERE

$\text{'}\text{'}\text{'}$if isinstance$($e$,$