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.

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.

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

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