The Gompertz Sigmoid Function and Its Derivative

Gompertz function - green; derivative - gray; logarithmic derivative - orange

upper asymptote a

3.69

growth rate b

-2.4

growth rate c

-1.825

filling

This Gompertz function is defined by

f(t)=a

b

ct

e

or

f(t)=aexp(bexp(ct))

, where

a

is the upper asymptote and

b

and

c

are the negative growth rates. The Gompertz function is a sigmoid function. It is a type of mathematical model for a time series, where growth is slowest at the start and end of a time period. This Demonstration plots the Gompertz function

f(t)

, its derivative,

f'(t)=abc

b

ct

e

ct

e

, and the ratio

d

dt

ln(f(t)

)=

f'(t)/f(t)

.