Sinusoidal Derivatives

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

Calculates the first and second derivatives of a sinusoidal function

Parameters

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

• second_constant (int or float) – Horizontal stretch factor of the original sine function; if zero, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

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

• fourth_constant (int or float) – Vertical shift of the original sine 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

sinusoidal_equation(), sinusoidal_integral(), sinusoidal_roots(), sinusoidal_model()

Notes

• Standard form of a sinusoidal function: \(f(x) = a\cdot{\sin(b\cdot(x - c))} + d\)

• First derivative of a sinusoidal function: \(f'(x) = ab\cdot{\cos(b\cdot(x - c))}\)

• Second derivative of a sinusoidal function: \(f''(x) = -ab^2\cdot{\sin(b\cdot(x - c))}\)

• Basic Differentiation Forumulas

• Chain Rule

• Derivatives of Trigonometric Functions

Examples

Import sinusoidal_derivatives function from regressions library

>>> from regressions.analyses.derivatives.sinusoidal import sinusoidal_derivatives

Generate the derivatives of a sinusoidal function with coefficients 2, 3, 5, and 7, then display the coefficients of its first and second derivatives

>>> derivatives_constants = sinusoidal_derivatives(2, 3, 5, 7)
>>> print(derivatives_constants['first']['constants'])
[6.0, 3.0, 5.0]
>>> print(derivatives_constants['second']['constants'])
[-18.0, 3.0, 5.0]

Generate the derivatives of a sinusoidal function with coefficients 7, -5, -3, and 2, then evaluate its first and second derivatives at 10

>>> derivatives_evaluation = sinusoidal_derivatives(7, -5, -3, 2)
>>> print(derivatives_evaluation['first']['evaluation'](10))
19.6859
>>> print(derivatives_evaluation['second']['evaluation'](10))
144.695

Generate the derivatives of a sinusoidal function with all inputs set to 0, then display the coefficients of its first and second derivatives

>>> derivatives_zeroes = sinusoidal_derivatives(0, 0, 0, 0)
>>> print(derivatives_zeroes['first']['constants'])
[0.0001, 0.0001, 0.0001]
>>> print(derivatives_zeroes['second']['constants'])
[-0.0001, 0.0001, 0.0001]