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” 