# 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).