Derivatives of Exponential Functions

One truly remarkable characteristic of ex is that

ex = ex    

Besides the trivial case of f (x) = 0, ex and its constant multiples are the only functions whose derivatives are equal to themselves!

Incorporating the principles of the chain rule, we might also say that if u is a function of x, then

eu = eu    

By extension, exdx = ex + c. Using the fact that

eu = eu    

we can derive a more general formula for the derivative of ax, where a is any positive constant.

First, note that ax can be rewritten as