# Hyperbolic tangent function

The ratio of subtraction of negative natural exponential function from positive natural exponential function to summation of positive and negative natural exponential functions is called the hyperbolic tangent function.

## Introduction

Assume, $x$ is a variable and $e$ is an irrational positive mathematical constant. The positive natural exponential function is written as $e^x$ and negative natural exponential function is written as $e^{-x}$.

The subtraction of negative natural exponential function from positive natural exponential function is $e^x\,–\,e^{-x}$.

The summation of positive and negative natural exponential functions is $e^x+e^{-x}$

The ratio of subtraction of them to summation of them can be expressed in mathematical form as follows.

$\large \dfrac{e^x-e^{-x}}{e^x+e^{-x}}$

The ratio is called the hyperbolic tangent function. The hyperbolic tangent is represented by $\tanh$ but the function is in terms of $x$. Hence, the hyperbolic tangent function is denoted by $\tanh{x}$ in mathematics.

$\large \tanh{x} = \dfrac{e^x-e^{-x}}{e^x+e^{-x}}$