
from math import factorial







class        lowercase	:




							def __init__(       self     ,A__     ,A__):
												lowercase					=		real
												if isinstance(A__     ,A__):
																	lowercase					=		[1] * rank
												else:
																	lowercase					=		rank




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




							def 				A__   (       self):
												lowercase					=		self.duals.copy()
												while cur[-1] == 0:
																	cur.pop(-1)
												return Dual(self.real     ,A__)




							def __add__(       self     ,A__):
												if not isinstance(A__     ,A__):
																	return Dual(self.real + other     ,self.duals)
												lowercase					=		self.duals.copy()
												lowercase					=		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__)))
												lowercase					=		[]
												for i in range(len(A__)):
																	new_duals.append(s_dual[i] + o_dual[i])
												return Dual(self.real + other.real     ,A__)

							lowercase_ : str       =__add__




							def __sub__(       self     ,A__):
												return self + other * -1




							def __mul__(       self     ,A__):
												if not isinstance(A__     ,A__):
																	lowercase					=		[]
																	for i in self.duals:
																						new_duals.append(i * other)
																	return Dual(self.real * other     ,A__)
												lowercase					=		[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__)

							lowercase_ : int       =__mul__




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




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





							def __pow__(       self     ,A__):
												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
												lowercase					=		self
												for _ in range(n - 1):
																	x *= self
												return x



def  UpperCamelCase					(							lowerCAmelCase__				,    lowerCAmelCase__				,    lowerCAmelCase__	):
					'''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'''	)
					lowercase					=		Dual(lowerCAmelCase__				,    1	)
					lowercase					=		func(lowerCAmelCase__	)
					if order == 0:
										return result.real
					return result.duals[order - 1] * factorial(lowerCAmelCase__	)


if __name__ == "__main__":
							import doctest

							doctest.testmod()



							def  UpperCamelCase					(							lowerCAmelCase__	):
												'''simple docstring'''
												return y**2 * y**4
							print(differentiate(f, 9, 2))






from numpy import exp, pi, sqrt



def  UpperCamelCase					(							lowerCAmelCase__				,    lowerCAmelCase__ = 0.0				,    lowerCAmelCase__ = 1.0	):
					'''simple docstring'''
					return 1 / sqrt(2 * pi * sigma**2	) * exp(-((x - mu) ** 2) / (2 * sigma**2)	)


if __name__ == "__main__":
							import doctest

							doctest.testmod()




