How it works ?
Recall that exponential functions are the only functions whose instantaneous rate of change (at any given input value) varies directly with the output of that function (evaluated at that particular input value).
Assuming there’s no carrying capacity or other inhibiting factors, the instantaneous growth rate of a population of a certain species at any time t depends solely upon the actual population at that time t. (Think about it: The instantaneous growth rate of a colony of bacteria at a time when there’s only 10 bacteria will be much lower than the instantaneous growth rate at the time when there’s 10,000 bacteria.)
Because of this, for any exponential function (that can be written of the form f(x)=ax, where a >0), its derivative will always be a scalar multiple of itself. For an informal illustration of this, see the applet below.
Interact with the applet for a few minutes, then answer the questions that follow.