Furthermore why is it that e^x is used in exponential
Furthermore why is it that is used in exponential modelling? Why aren't other exponential functions used, i.e. , etc?
Answer & Explanation
Apply the usual definition of the derivative:
Next, apply the usual laws of indices:
If we're able to show that as then we're done. This is easier said than done. We need to employ a little trickery. Let us first start by defining
so that , and hence . Finally, we have: . Hence:
This proof uses the definition that . To appreciate this definition, consider compound interest added yearly, monthly, weekly, daily, hourly, etc. Adding less and less interest, more and more frequently corresponds to letting t tends towards infinity in . Allowing a system to grow and decay on an infinitesimally small time scale gives rise to . This definition also benefits from a lack of circular reasoning: to be able to define as a power series required us knowing how to differentiate it and get its Taylor series. Either that, or it was an amazingly, infinitely-lucky guess.
If you define
then if you differentiate term by term you get back what you started with.
In some contexts it can be prudent to define the exponential function as the solution of the differential equation
so that the exponential function is it's own derivative because you define it as such.
By Picard's Existence Theorem the solution of this differential equation is unique. This seems like cheating but depending on what properties of you want to use it is perfectly valid. For example, this doesn't tell you how to even calculate without a little more work...
It is used in modelling because if you have a quantity that changes in proportion to it's size:
then the solution is . Lots of quantities change in proportion to their size; e.g. radioactive decay.