This section discusses the concept of exponential functions and their derivatives.

An exponential function is in the form of f(x) = a^x, where a is a positive constant. This function is defined in five segments:

1. When x = 0, f(x) = a⁰ = 1
   
2. When x is a positive integer n,
        
        f(x) = aⁿ = a·a·...a
   
   in which the total number of a is n.

1. When x is a negative integer -n,
        f(x) = a^-n = 1/aⁿ
   
2. When x is a rational number p/q in which p and q are integer and positive integer respectively,
        f(x) = a^p/q = a^p/a^q
   
3. When x is an irrational number,
        

Exponential function has some properties that are useful for calculating its value. They are given below.

1. If a > 0 and a ≠ 1, then the function f(x) = a^x is a continuous function in its domain and its value is positive.
   
2. If 0 < a < 1, the function f(x) = a^x is a decreasing function. However, if a > 1, the function is an increasing function.
   
3. a^x+y = a^xa^y
   
4. a^x-y = a^x/ a^y
   
5. (a^x)^y = a^xy
   
6. (ab)^x = a^xb^x
   
7. and
   when a > 1
   
8. and
   when 0 < a < 1

In some intervals, an exponential function is continuous and differentiable. The properties of the exponential function that relates its derivation is given in the next section.

If the exponential function, f(x) = a^x, is differentiable at 0, then it is differentiable everywhere in its interval and

     f(x) = f ^'(0)a^x

This equation means the rate of change of any exponential function is proportional to the function itself and this equation is exponential growth/decay equation. The increasing of the population is an example of exponential growth and the cooling of bodies is an exponential decay.

In the equation f(x) = f ^'(0)a^x, there must be an "a" such that f ^'(a) = 1. Traditionally, this "a" is denoted by the letter e, such that

The derivative of the exponential function f(x) = e^x is its function value which can be expressed as

     de^x/dx = e^x

The exponential function f(x) = e^x is an increasing continuous function in its domain. Its value is positive and

When plotted, the slope of a tangent line to the curve y = e^x is equal to the y coordination of that point as shown on the left. The Chain Rule can be apply to exponential function to facilitate the calculation of its derivative.

     de^u/dx = e^udu/dx

The exponential function can also be integrated and its value is

in which c is a constant.