from math import factorial
class  A_			:

  def __init__(		self			:   List[str]  ,				__SCREAMING_SNAKE_CASE			:   Any  ,				__SCREAMING_SNAKE_CASE			:   Optional[Any]							):
   __a							=	real
   if isinstance(A_  ,				A_							):
    __a							=	[1] * rank
   else:
    __a							=	rank

  def __repr__(		self			:   List[str]							):
   return (
       f"""{self.real}+"""
       f"""{"+".join(str(A_							)+"E"+str(n+1							)for n,dual in enumerate(self.duals							)							)}"""
   )

  def 		_UpperCAmelCase (		self			:   List[Any]							):
   __a							=	self.duals.copy()
   while cur[-1] == 0:
    cur.pop(-1							)
   return Dual(self.real  ,				A_							)

  def __add__(		self			:   Dict  ,				__SCREAMING_SNAKE_CASE			:   str							):




   if not isinstance(A_  ,				A_							):
    return Dual(self.real + other  ,				self.duals							)
   __a							=	self.duals.copy()
   __a							=	other.duals.copy()
   if len(A_							) > len(A_							):
    o_dual.extend([1] * (len(A_							) - len(A_							))							)
   elif len(A_							) < len(A_							):
    s_dual.extend([1] * (len(A_							) - len(A_							))							)
   __a							=	[]
   for i in range(len(A_							)							):
    new_duals.append(s_dual[i] + o_dual[i]							)
   return Dual(self.real + other.real  ,				A_							)

  _SCREAMING_SNAKE_CASE     					=      __add__

  def __sub__(		self			:   Union[str, Any]  ,				__SCREAMING_SNAKE_CASE			:   List[Any]							):
   return self + other * -1

  def __mul__(		self			:   int  ,				__SCREAMING_SNAKE_CASE			:   Any							):




   if not isinstance(A_  ,				A_							):
    __a							=	[]
    for i in self.duals:
     new_duals.append(i * other							)
    return Dual(self.real * other  ,				A_							)
   __a							=	[0] * (len(self.duals							) + len(other.duals							) + 1)
   for i, item in enumerate(self.duals							):
    for j, jtem in enumerate(other.duals							):
     new_duals[i + j + 1] += item * jtem
   for k in range(len(self.duals							)							):
    new_duals[k] += self.duals[k] * other.real
   for index in range(len(other.duals							)							):
    new_duals[index] += other.duals[index] * self.real
   return Dual(self.real * other.real  ,				A_							)

  _SCREAMING_SNAKE_CASE     					=      __mul__

  def __truediv__(		self			:   Optional[Any]  ,				__SCREAMING_SNAKE_CASE			:   Optional[Any]							):




   if not isinstance(A_  ,				A_							):
    __a							=	[]
    for i in self.duals:
     new_duals.append(i / other							)
    return Dual(self.real / other  ,				A_							)
   raise ValueError

  def __floordiv__(		self			:   Optional[Any]  ,				__SCREAMING_SNAKE_CASE			:   Any							):




   if not isinstance(A_  ,				A_							):
    __a							=	[]
    for i in self.duals:
     new_duals.append(i // other							)
    return Dual(self.real // other  ,				A_							)
   raise ValueError

  def __pow__(		self			:   List[str]  ,				__SCREAMING_SNAKE_CASE			:   List[Any]							):




   if n < 0 or isinstance(A_  ,				A_							):
    raise ValueError("power must be a positive integer"							)
   if n == 0:
    return 1
   if n == 1:
    return self
   __a							=	self
   for _ in range(n - 1							):
    x *= self
   return x



def 	__A		(      _A				,					_A				,					_A       ):



 """simple docstring"""


 if not callable(_lowerCAmelCase       ):
  raise ValueError("differentiate() requires a function as input for func"       )
 if not isinstance(_lowerCAmelCase				,					(float, int)       ):
  raise ValueError("differentiate() requires a float as input for position"       )
 if not isinstance(_lowerCAmelCase				,					_lowerCAmelCase       ):
  raise ValueError("differentiate() requires an int as input for order"       )
 __a							=	Dual(_lowerCAmelCase				,					1       )
 __a							=	func(_lowerCAmelCase       )
 if order == 0:
  return result.real
 return result.duals[order - 1] * factorial(_lowerCAmelCase       )


if __name__ == "__main__":
  import doctest

  doctest.testmod()



  def 	__A		(      _A       ):



   """simple docstring"""


   return y**2 * y**4






  print(differentiate(f, 9, 2))









def 						A_							(	_lowerCAmelCase				,  _lowerCAmelCase	)		->					Optional[int]:
 UpperCamelCase      :   List[Any]			      =  [1]
 for i in range(2				,  _lowerCAmelCase	):
  factorials.append(factorials[-1] * i	)
 assert 0 <= k < factorials[-1] * n, "k out of bounds"

 UpperCamelCase      :   Tuple			      =  []
 UpperCamelCase      :   str			      =  list(range(_lowerCAmelCase	)	)

 # Find permutation
 while factorials:
  UpperCamelCase      :   int			      =  factorials.pop()
  UpperCamelCase      ,       UpperCamelCase      :   int			      =  divmod(_lowerCAmelCase				,  _lowerCAmelCase	)
  permutation.append(elements[number]	)
  elements.remove(elements[number]	)
 permutation.append(elements[0]	)

 return permutation


if __name__ == "__main__":
    import doctest

    doctest.testmod()
