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



Then we have



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”