Exponential Derivatives

exponential_derivatives(first_constant, second_constant, precision=4)

Calculates the first and second derivatives of an exponential function

Parameters

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

• second_constant (int or float) – Base rate of variable of the original exponential 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

• 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

exponential_equation(), exponential_integral(), exponential_roots(), exponential_model()

Notes

• Standard form of an exponential function: \(f(x) = a\cdot{b^x}\)

• First derivative of an exponential function: \(f'(x) = a\cdot{\ln{b}\cdot{b^x}}\)

• Second derivative of an exponential function: \(f''(x) = a\cdot{\ln^2{b}\cdot{b^x}}\)

• Basic Differentiation Forumulas

• Derivatives of Exponential Functions

Examples

Import exponential_derivatives function from regressions library

>>> from regressions.analyses.derivatives.exponential import exponential_derivatives

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

>>> derivatives_constants = exponential_derivatives(2, 3)
>>> print(derivatives_constants['first']['constants'])
[2.1972, 3.0]
>>> print(derivatives_constants['second']['constants'])
[2.4139, 3.0]

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

>>> derivatives_evaluation = exponential_derivatives(-2, 3)
>>> print(derivatives_evaluation['first']['evaluation'](10))
-129742.4628
>>> print(derivatives_evaluation['second']['evaluation'](10))
-142538.3811

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

>>> derivatives_zeroes = exponential_derivatives(0, 0)
>>> print(derivatives_zeroes['first']['constants'])
[-0.0009, 0.0001]
>>> print(derivatives_zeroes['second']['constants'])
[0.0083, 0.0001]