#! /usr/bin/env python3
#
def newton_example ( ):

  from sympy import cbrt
  from sympy import diff
  from sympy import exp
  from sympy import symbols
  import sympy

  print ( '' )
  print ( 'newton_example():' )
  print ( '  sympy version: ' + sympy.__version__ )
  print ( '  We plan to apply the Newton method to a function f(x).' )
  print ( '  We have f(x) = x^(1/3) exp(-x^2).' )
  print ( '  We want a formula for the derivative of f(x).' )
  print ( '  We also seek the second derivative.' )

  x = symbols ( 'x' )

  print ( '' )
  print ( '  Initially, we define f(x) using a fractional power.' )
  print ( '  sympy() treats 1/3 as a numerical value, not a symbolic.' )
  print ( '' )
  print ( '  f = x**(1/3) * exp(-x**2)' )

  f = x**(1/3) * exp ( - x**2 )

  dfdx = diff ( f, x )
  print ( '  d f /dx = ', dfdx )

  d2fdx2 = diff ( f, x, 2 )
  print ( '  d2 f /dx2 = ', d2fdx2 )

  print ( '' )
  print ( '  Redefine f(x) using the sympy() cbrt() function.' )
  print ( '  Now we expect a purely symbolic result.' )
  print ( '' )
  print ( '  f = cbrt(x) * exp(-x**2)' )

  f = cbrt ( x ) * exp ( - x**2 )

  print ( '' )
  print ( '  f = cbrt(x) - exp(-x**2)' )

  dfdx = diff ( f, x )
  print ( '  d f /dx = ', dfdx )

  d2fdx2 = diff ( f, x, 2 )
  print ( '  d2 f /dx2 = ', d2fdx2 )

  return

if ( __name__ == "__main__" ):
  newton_example ( )