# Exponential Function

## Function

`exp(x)`

## Summary

Returns the result of the constant e raised to the x power. The exponential function is the inverse of the natural logarithm function.

## Input

Name | Description |
---|---|

x | the exponent part of the expression |

## Output

## Explanation

The output of the exponential function is proportional to the rate of change of the function. For example, the output of the function at is equal to and the rate of change of the function at is also equal to the output . In other words, the slope of the function at any given point is equal to the value of the function at that point.

Conceptually, the function can be defined in the form , where the constant e is a number approximately equal to . This number and the function are directly related to the concept of exponential growth.

### Inverse of the Exponential Function

The natural logarithm function is the inverse of the exponential function. This can be visually represented by the graph of the two functions, where the image of one function is the reflection of the other function over the diagonal line.

Just as the exponential function is related to the concept of exponential growth, the natural logarithm function is related to the inverse concept: exponential decay. Here are some simple examples of the input and output being reversed.

### Derivative of the Exponential Function

In calculus, the derivative of the exponential function has the unique property of being the itself. This is because the output of the function grows proportionally to its current value.

We can demonstrate this fact using the power series definition of the exponential function shown below.

When we take the derivative of the expressions of the power series the first term goes to 0, the second term simplifies to what used to be the first term, the third term simplifies to what used to be the second term and so on.

## Definition

The exponential function can be defined in terms of the power series given below, which as the number of terms approaches infinity approaches the function.