This example shows how to verify the properties of the exponential function using the power series definition of the function shown below:

The properties of the exponential function verified in this example are summarized in the table below.

Property | Equation |
---|---|

Zero Property | |

Derivative Property | |

Inverse Property | |

Addition Property* | |

Subtraction Property* | |

Multiplication Property* | |

General Exponentiation* |

*Coming soon.

The zero property can be easily verified by plugging zero into the power series definition of the exponential function.

Start with the power series definition of the exponential function.

Substitute in for .

All of the terms besides the first term go to zero.

This verifies the zero property.

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 power series definition of the exponential function.

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.

Take the derivative of each polynomial.

Applying the identity property of addition we can remove the leading , leaving an expression identical to the original power series definition.

Substituting the exponential function back into the expression we see that the derivative of the exponential function is equal to itself.

This verifies the derivative property of the exponential function.

This example verifies the inverse property of the exponential function using the distributive property of multiplication.

Start with the inverse property of the exponential function.

Multiply both sides by and rearrange the equation so is on the left side.

Apply the power series definition of the exponential function, shown below, to both instances of the exponential function from above.

After substituting the power series definition we are left with the following expression.

Note, the expressions where is raised to an odd power will flip the sign to negative and expressions where is raised to an even power the sign will stay positive. This is shown below.

Then, we can use a geometric interpretation of the distributive property of multiplication to evaluate the product of the two series. This is represented below as the area of the rectangle.

This geometric interpretation of the multiplication is illustrated in the figure below. The area of the whole is equal to all of the sub-areas summed together.

The summation of the geometric interpretation of multiplication is shown below.

If we group the same degree polynomials and sum together all of the terms, everything else goes to zero!

Since and all of the other polynomials sum to this verifies the inverse property.