Recall that exponential functions are the only functions whose [color=#cc0000][b]instantaneous rate of change[/b][/color] (at any given input value) varies directly with the output of that function (evaluated at that particular input value). [br][br][color=#1e84cc][b]For example: [/b][/color] [br][i][color=#1e84cc]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.) [/color][/i][br][br]Because of this, [b]for any exponential function (that can be written of the form [math]f\left(x\right)=a^x[/math][/b][b], where a >0),[/b] [color=#cc0000][b]its derivative will always be a scalar multiple of itself. [/b][/color] For an informal illustration of this, see the applet below. [br][br]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 [math]f\left(x\right)=a^x[/math], then [math]f'\left(x\right)=k\cdot f\left(x\right)[/math], where k is a scalar.
Here's a BIG HINT -- You'll formally prove this fact true in a few weeks:[br][br]limit (h -->0) of [math]\frac{a^h-1}{h}[/math] is equal to [math]ln\left(a\right)[/math].
For the previous question, what is the exact value of this constant (scalar) k?
[math]k=ln\left(a\right)[/math]
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).