# When does f'(x) = f(x)?

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). For example: 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 , 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.

Is there a certain base value of a for which f'(x) = f(x) for all x? If so, what do you think this particular base value is?

Use the formal limit definition of a derivative to show that if , then , where k is a scalar.

For the previous question, what is the exact value of this constant (scalar) k?

Use your results from the previous two questions to formally prove that f'(x) = f(x) for all x in the domain of f when k = the value you provided in response to question (1).