The inverse of the hyperbolic cosecant function, $\text{csch}(x)$, can be derived using the following process:

Step-by-Step Solution for Finding the Inverse of $\text{csch}(x)$

We start with the equation:

$y = \text{csch}(x)$

The hyperbolic cosecant is defined as:

$y = \frac{1}{\sinh(x)}$

Now, solve for $x$ in terms of $y$:

$y = \frac{1}{\sinh(x)} \implies \sinh(x) = \frac{1}{y}$

Next, recall that the inverse of $\sinh(x)$ is $\text{arsinh}(x)$, which is expressed as:

$x = \text{arsinh}\left(\frac{1}{y}\right)$

The formula for the inverse hyperbolic sine (arsinh) is:

$\text{arsinh}(x) = \ln(x + \sqrt{x^2 + 1})$

So, we substitute $\frac{1}{y}$ into the inverse hyperbolic sine formula:

$x = \ln\left(\frac{1}{y} + \sqrt{\left(\frac{1}{y}\right)^2 + 1}\right)$

Simplify the expression:

$x = \ln\left(\frac{1}{y} + \sqrt{\frac{1}{y^2} + 1}\right)$

$x = \ln\left(\frac{1}{y} + \frac{\sqrt{1 + y^2}}{|y|}\right)$

This gives the inverse of the hyperbolic cosecant function:

$\text{csch}^{-1}(y) = \ln\left(\frac{1}{y} + \frac{\sqrt{1 + y^2}}{|y|}\right)$

Thus, the inverse of $\text{csch}(x)$ is:

$\boxed{\text{csch}^{-1}(x) = \ln\left(\frac{1}{x} + \frac{\sqrt{1 + x^2}}{|x|}\right)}$

Let me know if you'd like further details!

Related Questions:

1. What is the inverse of the hyperbolic cosine $\cosh^{-1}(x)$?
2. How do you derive the inverse of $\sinh(x)$?
3. What are the properties of the hyperbolic functions?
4. Can you solve an equation involving $\text{csch}(x)$ and its inverse?
5. What are the applications of hyperbolic functions in physics?

Tip: The logarithmic form of inverse hyperbolic functions makes them useful in solving integrals and differential equations involving hyperbolic terms.