# Derivative of inverse hyperbolic functions

• mvantuyl
In summary, taking the derivative of inverse hyperbolic functions, such as sinh^{-1}(x), involves first rewriting the function in terms of its original variable and then differentiating.
mvantuyl

## Homework Statement

I don't understand how to take the derivative of inverse hyperbolic functions such as sinh$$^{-1}$$(x). I know that the derivative of sinh(x) is cosh(x) but don't know what to do with the inverse.

## The Attempt at a Solution

I'm completely at a loss here. Could somebody point me in the right direction? (I don't necessarily want the answer, just a shove along the path would be wonderful)

Welcome to PF!

Hi mvantuyl! Welcome to PF!

Hint: if y = sinh-1x, then write x = sinhy, and then differentiate (and then convert back into x's, of course)!

Thank you! It's so obvious I managed to completely overlook it. :)

## What is the derivative of the inverse hyperbolic function?

The derivative of the inverse hyperbolic function is the reciprocal of the derivative of the corresponding hyperbolic function. For example, the derivative of the inverse hyperbolic cosine function (arccosh) is 1/sqrt(x^2 - 1).

## What is the general formula for finding the derivative of inverse hyperbolic functions?

The general formula for finding the derivative of inverse hyperbolic functions is: d/dx(arccosh(x)) = 1/sqrt(x^2 - 1), d/dx(arcsinh(x)) = 1/sqrt(x^2 + 1), and d/dx(arctanh(x)) = 1/(1 - x^2).

## How do you use the chain rule to find the derivative of inverse hyperbolic functions?

The chain rule can be used to find the derivative of inverse hyperbolic functions by first rewriting the function in terms of the corresponding hyperbolic function and then using the chain rule on the hyperbolic function.

## What is the relationship between the derivative of inverse hyperbolic functions and the derivative of their corresponding hyperbolic functions?

The derivative of inverse hyperbolic functions are related to the derivative of their corresponding hyperbolic functions through the chain rule. The derivative of the inverse hyperbolic function is the reciprocal of the derivative of the corresponding hyperbolic function.

## Are there any special cases for finding the derivative of inverse hyperbolic functions?

Yes, there are some special cases when finding the derivative of inverse hyperbolic functions. For example, the derivative of the inverse hyperbolic tangent function (arctanh) is 1/(1 - x^2), but when x = 1 or -1, the derivative is undefined. Similarly, for the inverse hyperbolic secant function (arcsech), the derivative is undefined when x = 0 or 1.

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