The hyperbolic cosine function is a mathematical function that is closely associated with exponential growth.

The hyperbolic cosine function, denoted as $\cosh(x)$, is defined by the formula:

This function is classified as an even function, which means it satisfies the property $\cosh(-x) = \cosh(x)$. The range of $\cosh(x)$ consists of all positive real numbers.

The graph of $\cosh(x)$ is a smooth, U-shaped curve. As $x$ approaches infinity, the value of $\cosh(x)$ also approaches infinity. Conversely, as $x$ approaches 0, $\cosh(x)$ approaches 1. This function is frequently utilized in mathematical modeling to depict scenarios involving exponential growth, such as population dynamics or the spread of diseases.

The derivative of $\cosh(x)$ is the hyperbolic sine function, denoted as $\sinh(x)$. The hyperbolic sine function is defined by the expression:

Using the chain rule, we can compute the derivative of $\cosh(x)$ as follows:

The inverse hyperbolic cosine function, denoted as $\cosh^{-1}(x)$, serves as the inverse of $\cosh(x)$ and is defined by the formula:

The domain of $\cosh^{-1}(x)$ includes all real numbers greater than or equal to $1$, while its range encompasses all real numbers.

In summary, the hyperbolic cosine function is a valuable tool in mathematical modeling and finds numerous applications across various fields of science and engineering.