This example demonstrates how to derive the power series definition of the exponential function, shown below, using a taylor series approximation.

This example uses two properties of the exponential function. The first is that the output of the function at is . And the second is that the derivative property.

Start with the general form of a taylor series expansion, where the function is exponential function .

Choose a point to expand the series around. In the case of the exponential function the choice of is a likely choice because the output is . Substitute into the series.

Simplify.

Find the first, second, third, etc. derivatives of the function.

Calculate the output given the input of , recall .

Substitute these values into the series.

Replace the generic name with the abbreviated name and this gives the power series of the exponential function. Optionally, expand the factorial operator in the denominator.

The exponential function models exponential growth. The output of the function at any given point is equal to the rate of change at that point. For real number input, the function conceptually returns Euler's number raised to the value of the input.

A taylor series is a tool in mathematics to define a function in terms of an infinite power series.