Verify Exponential Function Properties
This example shows how to verify the properties of the exponential function using the power series definition of the function. The power series definition used throughout this example is given by the formula below. This definition can be derived using a Taylor Series^{[1]}.
Summarized in the table below are the properties of the exponential function.
Property  Equation 

Zero Property  
Derivative Property  
Inverse Property  
Addition Property  
Subtraction Property  
Multiplication Property  
General Exponentiation 
Verify Zero Property of Exponential Function
The zero property can be easily verified by pluggin zero into the power series definition of the exponential function.

Start with the formal definition.
Substitue in for .

Since all of the terms besides the first term go to zero, we have verified the zero property.
Verify Derivative Property of Exponential Function
The derivative property of the exponential function can be verified by taking the derivative of the power series definition of the function. Since the power series is a summation, the derivative of each of the terms can be found individually.

Start with the formal definition.
Substitue in for .

Take the derivative of both sides.

Since the power series is made up of a bunch of summations, the derivative can be applied individually to each term.
Finding the derivative of each term yields the following pattern.
Here we can see the uniqueness of the exponential function. The derivative of the function is equal to itself.
Verify Inverse Property of Exponential Function
The inverse property of the exponential function is verified by applying a geometric interpreatation to the distributive property of multiplication.

Start with the claim of the inverse property.
Multiply both sides by .

Apply the power series definition of the exponential function to both and .

Draw the geometric interpretation of the distributive property of multiplication for the product of the two power series.
Then when the terms are grouped together, we see that everything besides the first term goes to .
Since this verifies the inverse property.