Derivatives of Exponential Functions
One truly remarkable characteristic of e^{x} is that
Besides the trivial case of f (x) = 0, e^{x} 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
e^{u} = e^{u} 

By extension, e^{x}dx = e^{x} + c. Using the fact that
e^{u} = e^{u} 

we can derive a more general formula for the derivative of a^{x}, where a is any
positive constant.
First, note that a^{x} can be rewritten as