Derivative of an Exponential

 

An important result in Calculus is the rate of change of the exponential function

 

           

 

As a first step we define a change in “y” for a small change in “x” as

 

 

And the slope is

 

 

Considering the definition of a logarithm, we can write in general

 

 

or specifically

 

 

Using a result for the derivation of the derivative of the logarithm we can express “e” as a limit as follows

 

 

and if we set  then

 

Combining everything, we can apply the differential operator as follows

 

 

Where we can simplify the new variable further, as

 

 

And we can use a particular path for achieving the limit of (Δx, h) -> (0,0) by setting , as  so that

 

 

Note that this result would be the same if we used any other arbitrary path as for example, , as   so that

 

 

Although this is more difficult to verify, one could put this on a spreadsheet to give confidence in the result even if this was not mathematically rigorous.

 

In any event the final result is

 

 

or for the special case when the base is the constant “e”