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Limits Regarding LN and the Derivative of Exponential Functions


Part One. Proving Important Limits

In part one I seek to explain/prove the following limit as eloquently as possible.


You can easily convince yourself that this is true by testing the limit numerically by choosing any a and plugging in h-values increasingly close to 0. However, we will use a more mathematical approach. 

Why is this important?  It turns out it is needed when finding the derivative of exponential functions using the definition of the derivative.

In order to begin the proof, you must recall two very important and identical limits from calculus (the Calculus based definition of "e" called Napier's constant or Euler's number).

   

Before we begin on the generic case, let's prove the specific case where a=e -> ln(e)=1.


We will start with the right hand side. In this example, it is helpful to make substitution by letting the numerator be equal to h. That is:


Thus we can change our limit into terms of h. Note that as x->0, h->0 because e^0-1=1-1=0.


Now with a little bit of algebra (reciprocal rules, log laws, and limit laws) we get:


You may notice that the limit within the natural log function to be that definition of e.


QED. In summation:


So this specific case when a=e is confirmed. However, what about the general case? It turns out that we will need this result to show the general case from the very beginning. 


Again, let's start with the right hand side. We will first use the fact that e^ln(x)=x (cancellation principle between inverse functions).
 

Now, if we multiply the top and bottom by ln(a) and make another variable change x*ln(a)=h. Again, as x->0, h->0.


And now recognize the remaining limit which we just proved to be 1.


QED! In summation:



What should you takeaway from part one?

The following limit formulas are always true and sometimes useful in calculus.

   




Part Two. Using the Definition to find the Derivative of Exponential Functions

Let's begin by finding the derivative of the natural exponential function. We will be using the limits we proved in part one.


Now recall the definition of the derivative. We will use it to find f'(x).


You may notice a common factor of e^x in the numerator. This can be factored and pass through the limit since x does not depend on h. You will notice we can use the limit we proved in part one.


Thus showing one of the most interesting derivatives in all of Calculus, the derivative of e^x is itself (and the only function which exhibits this trait).


Now, to generalize and analyze the derivative of any based exponential function.


Again, following the difference quotient, the proof is nearly identical, except that the limit will not turn into 1 but instead into the ln(a) because of part one.


Or in summary:


It's actually quite simple in comparison to part 1. Note that you can prove this derivative quite easily without these limits using implicit differentiation but I enjoy this approach as it uses the definition of the derivative.


What should you takeaway from part two?

The derivative of the natural exponential function and an exponential functions of any base is:

    

Memorize these derivatives :)


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