If  and , then the exponential function with base  is given by .

Natural Exponential Functions

1.  The irrational number    or .

2.  For each real , .

3.   is a natural exponential function.

Derivatives of Exponential Functions

1..

2..

The hyperbolic cosine and sine functions

The hyperbolic cosine function is defined as

,

while the hyperbolic sine function is defined as 

.

Properties of hyperbolic cosine and sine functions

1. .

2. .

3.