Some Definitions of e

1.

2.   e is the unique +ve number for which

3.

The second one is the significant one for us since that limit is exactly the limit which we're working with above.  Thus, this definition leads to the following fact,

Fact 1

For the natural exponential function, f ( x ) = e^x  we have

Hence, provided we are using the natural exponential function we obtain the following.

                                      f ( x )= e^x       ⇒                f ′ ( x ) = e^x

At this instance we're missing some knowledge that will let us to simply get the derivative for a general function.  We will be able to show that eventually for a general exponential function we have,

                            f ( x ) = a ^x   ⇒       f ′ ( x ) = a ^x ln ( a )