from sympy import *
init_printing()  # Set up fancy printing

import math
math.sqrt(2)

sqrt(2)  # This `sqrt` comes from SymPy

cos(0)

# Call acos on -1 to find where on the circle the x coordinate equals -1



# Call `math.acos` on -1 to find the same result using the builtin math module.  
# Is the result the same?  
# What do you think `numpy.acos` give you?



x, y, z = symbols('x,y,z')
alpha, beta, gamma = symbols('alpha,beta,gamma')

x + 1

log(alpha ** beta) + gamma

sin(x)**2 + cos(x)**2

?, ? = symbols('?')

exp(?)

(x**2).diff(x)

sin(x).diff(x)

(x**2 + x*y + y**2).diff(x)

(x**2 + x*y + y**2).diff(y)

mu, sigma = symbols('mu,sigma')

bell = exp((x - mu)**2 / sigma**2)
bell

?.diff(?)

# Derivative of bell curve with respect to sigma



#  Find the second and third derivative of `bell`



expr = sin(x)**2 + cos(x)**2
expr

simplify(expr)

bell.diff(x).diff(x).diff(x)

# Call simplify on the third derivative of the bell expression



sympify('r * cos(theta)^2')