
from math import factorial




class 						lowercase__  :
	def __init__(			self							:   Optional[int]       ,   UpperCAmelCase_							:   List[str]       ,   UpperCAmelCase_							:   int							):
						SCREAMING_SNAKE_CASE__													= real
						if isinstance(UpperCAmelCase_       ,   UpperCAmelCase_							):
											SCREAMING_SNAKE_CASE__													= [1] * rank
						else:
											SCREAMING_SNAKE_CASE__													= rank
	def __repr__(			self							:   Optional[int]							):
						return (
						    F'{self.real}+'
						    F'{"+".join(str(UpperCAmelCase_							)+"E"+str(n+1							)for n,dual in enumerate(self.duals							)							)}'
						)
	def      A_				(			self							:   Optional[int]							):
						SCREAMING_SNAKE_CASE__													= self.duals.copy()
						while cur[-1] == 0:
											cur.pop(-1							)
						return Dual(self.real       ,   UpperCAmelCase_							)
	def __add__(			self							:   List[Any]       ,   UpperCAmelCase_							:   Optional[int]							):
						if not isinstance(UpperCAmelCase_       ,   UpperCAmelCase_							):
											return Dual(self.real + other       ,   self.duals							)
						SCREAMING_SNAKE_CASE__													= self.duals.copy()
						SCREAMING_SNAKE_CASE__													= other.duals.copy()
						if len(UpperCAmelCase_							) > len(UpperCAmelCase_							):
											o_dual.extend([1] * (len(UpperCAmelCase_							) - len(UpperCAmelCase_							))							)
						elif len(UpperCAmelCase_							) < len(UpperCAmelCase_							):
											s_dual.extend([1] * (len(UpperCAmelCase_							) - len(UpperCAmelCase_							))							)
						SCREAMING_SNAKE_CASE__													= []
						for i in range(len(UpperCAmelCase_							)							):
											new_duals.append(s_dual[i] + o_dual[i]							)
						return Dual(self.real + other.real       ,   UpperCAmelCase_							)

	A__    :   str				       =__add__
	def __sub__(			self							:   List[str]       ,   UpperCAmelCase_							:   List[str]							):
						return self + other * -1
	def __mul__(			self							:   Optional[Any]       ,   UpperCAmelCase_							:   List[str]							):
						if not isinstance(UpperCAmelCase_       ,   UpperCAmelCase_							):
											SCREAMING_SNAKE_CASE__													= []
											for i in self.duals:
																new_duals.append(i * other							)
											return Dual(self.real * other       ,   UpperCAmelCase_							)
						SCREAMING_SNAKE_CASE__													= [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       ,   UpperCAmelCase_							)

	A__    :   Union[str, Any]				       =__mul__
	def __truediv__(			self							:   List[str]       ,   UpperCAmelCase_							:   Optional[Any]							):
						if not isinstance(UpperCAmelCase_       ,   UpperCAmelCase_							):
											SCREAMING_SNAKE_CASE__													= []
											for i in self.duals:
																new_duals.append(i / other							)
											return Dual(self.real / other       ,   UpperCAmelCase_							)
						raise ValueError
	def __floordiv__(			self							:   Tuple       ,   UpperCAmelCase_							:   Optional[int]							):
						if not isinstance(UpperCAmelCase_       ,   UpperCAmelCase_							):
											SCREAMING_SNAKE_CASE__													= []
											for i in self.duals:
																new_duals.append(i // other							)
											return Dual(self.real // other       ,   UpperCAmelCase_							)
						raise ValueError





	def __pow__(			self							:   List[str]       ,   UpperCAmelCase_							:   List[str]							):
						if n < 0 or isinstance(UpperCAmelCase_       ,   UpperCAmelCase_							):
											raise ValueError('power must be a positive integer'							)
						if n == 0:
											return 1
						if n == 1:
											return self
						SCREAMING_SNAKE_CASE__													= self
						for _ in range(n - 1							):
											x *= self
						return x






def 							_lowercase							(     UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_		)		->		str:
					'''simple docstring'''


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


if __name__ == "__main__":
				import doctest

				doctest.testmod()






				def 							_lowercase							(     UpperCamelCase_		)		->		Tuple:
									'''simple docstring'''


									return y**2 * y**4



				print(differentiate(f, 9, 2))







def 							_lowercase							(     UpperCamelCase_	,	UpperCamelCase_		)		->		str:
					'''simple docstring'''


					if not isinstance(UpperCamelCase_	,	UpperCamelCase_		):
										raise ValueError('iterations must be defined as integers'		)
					if not isinstance(UpperCamelCase_	,	UpperCamelCase_		) or not number >= 1:
										raise ValueError(
										    'starting number must be\n                         and integer and be more than 0'		)
					if not iterations >= 1:
										raise ValueError('Iterations must be done more than 0 times to play FizzBuzz'		)

					SCREAMING_SNAKE_CASE__													= ''
					while number <= iterations:
										if number % 3 == 0:
															out += "Fizz"
										if number % 5 == 0:
															out += "Buzz"
										if 0 not in (number % 3, number % 5):
															out += str(UpperCamelCase_		)

										# print(out)
										number += 1
										out += " "
					return out


if __name__ == "__main__":
				import doctest

				doctest.testmod()





