




"""simple docstring"""






def      _snake_case			(				lowercase__							,    lowercase__		):
				# "extended trapezoidal rule"
				# int(f) = dx/2 * (f1 + 2f2 + ... + fn)
				_lowerCamelCase     :			List[Any]    =    (boundary[1] - boundary[0]) / steps
				_lowerCamelCase     :			Tuple    =    boundary[0]
				_lowerCamelCase     :			Dict    =    boundary[1]
				_lowerCamelCase     :			List[Any]    =    make_points(lowercase__							,    lowercase__							,    lowercase__		)
				_lowerCamelCase     :			List[Any]    =    0.0
				y += (h / 2.0) * f(lowercase__		)
				for i in x_i:
								# print(i)
								y += h * f(lowercase__		)
				y += (h / 2.0) * f(lowercase__		)
				return y







def      _snake_case			(				lowercase__							,    lowercase__							,    lowercase__		):
				_lowerCamelCase     :			str    =    a + h
				while x < (b - h):
								yield x
								_lowerCamelCase     :			int    =    x + h







def      _snake_case			(				lowercase__		):  # enter your function here
				_lowerCamelCase     :			Optional[Any]    =    (x - 0) * (x - 0)
				return y







def      _snake_case			(				):
				_lowerCamelCase     :			int    =    0.0  # Lower bound of integration
				_lowerCamelCase     :			Optional[int]    =    1.0  # Upper bound of integration
				_lowerCamelCase     :			List[str]    =    1_0.0  # define number of steps or resolution
				_lowerCamelCase     :			List[Any]    =    [a, b]  # define boundary of integration
				_lowerCamelCase     :			Optional[Any]    =    method_a(lowercase__							,    lowercase__		)
				print(f'''y = {y}'''		)


if __name__ == "__main__":
			main()







from math import factorial





class 			__a  :







 def __init__(		self					:   Union[str, Any]							,   SCREAMING_SNAKE_CASE					:   Tuple							,   SCREAMING_SNAKE_CASE					:   Optional[int]						):


      '''simple docstring'''






      UpperCamelCase__			:   Tuple       				=  real
      if isinstance(SCREAMING_SNAKE_CASE							,   SCREAMING_SNAKE_CASE						):
           UpperCamelCase__			:   Union[str, Any]       				=  [1] * rank
      else:
           UpperCamelCase__			:   int       				=  rank







 def __repr__(		self					:   Tuple						):


      '''simple docstring'''






      return (
          F'{self.real}+'
          F'{"+".join(str(SCREAMING_SNAKE_CASE						)+"E"+str(n+1						)for n,dual in enumerate(self.duals						)						)}'
      )







 def 					__lowercase					(		self					:   Tuple						):


      '''simple docstring'''






      UpperCamelCase__			:   Optional[Any]       				=  self.duals.copy()
      while cur[-1] == 0:
           cur.pop(-1						)
      return Dual(self.real							,   SCREAMING_SNAKE_CASE						)







 def __add__(		self					:   Union[str, Any]							,   SCREAMING_SNAKE_CASE					:   Dict						):


      '''simple docstring'''






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

 _lowerCAmelCase       :       Dict        = __add__







 def __sub__(		self					:   Tuple							,   SCREAMING_SNAKE_CASE					:   Tuple						):


      '''simple docstring'''






      return self + other * -1







 def __mul__(		self					:   int							,   SCREAMING_SNAKE_CASE					:   Dict						):


      '''simple docstring'''






      if not isinstance(SCREAMING_SNAKE_CASE							,   SCREAMING_SNAKE_CASE						):
           UpperCamelCase__			:   str       				=  []
           for i in self.duals:
                new_duals.append(i * other						)
           return Dual(self.real * other							,   SCREAMING_SNAKE_CASE						)
      UpperCamelCase__			:   List[Any]       				=  [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							,   SCREAMING_SNAKE_CASE						)

 _lowerCAmelCase       :       Union[str, Any]        = __mul__







 def __truediv__(		self					:   List[Any]							,   SCREAMING_SNAKE_CASE					:   Any						):


      '''simple docstring'''






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







 def __floordiv__(		self					:   Union[str, Any]							,   SCREAMING_SNAKE_CASE					:   Tuple						):


      '''simple docstring'''






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







 def __pow__(		self					:   str							,   SCREAMING_SNAKE_CASE					:   int						):


      '''simple docstring'''






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







def 		SCREAMING_SNAKE_CASE							(					__lowerCAmelCase					,    __lowerCAmelCase					,    __lowerCAmelCase			) ->	str:
     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"			)
     UpperCamelCase__			:   Optional[Any]       				=  Dual(__lowerCAmelCase					,    1			)
     UpperCamelCase__			:   Any       				=  func(__lowerCAmelCase			)
     if order == 0:
          return result.real
     return result.duals[order - 1] * factorial(__lowerCAmelCase			)


if __name__ == "__main__":
 import doctest

 doctest.testmod()







 def 		SCREAMING_SNAKE_CASE							(					__lowerCAmelCase			) ->	List[Any]:
      return y**2 * y**4




 print(differentiate(f, 9, 2))