Logarithmic Derivatives

logarithmic_derivatives(first_constant, second_constant, precision=4)

Calculates the first and second derivatives of a logarithmic function

Parameters

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

• 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; if zero inputted as argument, it will be converted to a small, non-zero decimal value (e.g., 0.0001)

See also

logarithmic_equation(), logarithmic_integral(), logarithmic_roots(), logarithmic_model()

Notes

• Standard form of a logarithmic function: \(f(x) = a\cdot{\ln{x}} + b\)

• First derivative of a logarithmic function: \(f'(x) = a\cdot{\frac{1}{x}}\)

• Second derivative of a logarithmic function: \(f''(x) = -a\cdot{\frac{1}{x^2}}\)

• Basic Differentiation Forumulas

• Derivatives of Logarithmic Functions

Examples

Import logarithmic_derivatives function from regressions library

>>> from regressions.analyses.derivatives.logarithmic import logarithmic_derivatives

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

>>> derivatives_constants = logarithmic_derivatives(2, 3)
>>> print(derivatives_constants['first']['constants'])
[2.0]
>>> print(derivatives_constants['second']['constants'])
[-2.0]

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

>>> derivatives_evaluation = logarithmic_derivatives(-2, 3)
>>> print(derivatives_evaluation['first']['evaluation'](10))
-0.2
>>> print(derivatives_evaluation['second']['evaluation'](10))
0.02

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

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