Cubic Derivatives

cubic_derivatives(first_constant, second_constant, third_constant, fourth_constant, precision=4)

Calculates the first and second derivatives of a cubic function

Parameters

  • first_constant (int or float) – Coefficient of the cubic term of the original cubic function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

  • second_constant (int or float) – Coefficient of the quadratic term of the original cubic function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

  • third_constant (int or float) – Coefficient of the linear term of the original cubic function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

  • fourth_constant (int or float) – Coefficient of the constant term of the original cubic function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

  • precision (int, default=4) – Maximum number of digits that can appear after the decimal place of the resultant roots

Raises

  • TypeError – First four arguments must be integers or floats

  • ValueError – Last argument must be a positive integer

Returns

  • derivatives[‘first’][‘constants’] (list of float) – Coefficients of the resultant first derivative

  • derivatives[‘first’][‘evaluation’] (func) – Function for evaluating the resultant first derivative at any float or integer argument

  • derivatives[‘second’][‘constants’] (list of float) – Coefficients of the resultant second derivative

  • derivatives[‘second’][‘evaluation’] (func) – Function for evaluating the resultant second derivative at any float or integer argument

See also

cubic_equation(), cubic_integral(), cubic_roots(), cubic_model()

Notes

  • Standard form of a cubic function: \(f(x) = a\cdot{x^3} + b\cdot{x^2} + c\cdot{x} + d\)

  • First derivative of a cubic function: \(f'(x) = 3a\cdot{x^2} + 2b\cdot{x} + c\)

  • Second derivative of a cubic function: \(f''(x) = 6a\cdot{x} + 2b\)

  • Basic Differentiation Forumulas

Examples

Import cubic_derivatives function from regressions library
>>> from regressions.analyses.derivatives.cubic import cubic_derivatives
Generate the derivatives of a cubic function with coefficients 2, 3, 5, and 7, then display the coefficients of its first and second derivatives
>>> derivatives_constants = cubic_derivatives(2, 3, 5, 7)
>>> print(derivatives_constants['first']['constants'])
[6.0, 6.0, 5.0]
>>> print(derivatives_constants['second']['constants'])
[12.0, 6.0]
Generate the derivatives of a cubic function with coefficients 7, -5, -3, and 2, then evaluate its first and second derivatives at 10
>>> derivatives_evaluation = cubic_derivatives(7, -5, -3, 2)
>>> print(derivatives_evaluation['first']['evaluation'](10))
1997.0
>>> print(derivatives_evaluation['second']['evaluation'](10))
410.0
Generate the derivatives of a cubic function with all inputs set to 0, then display the coefficients of its first and second derivatives
>>> derivatives_zeroes = cubic_derivatives(0, 0, 0, 0)
>>> print(derivatives_zeroes['first']['constants'])
[0.0003, 0.0002, 0.0001]
>>> print(derivatives_zeroes['second']['constants'])
[0.0006, 0.0002]