Day 37

 The Derevative of the Inverse of Hyperbolic 

function

The six inverse hyperbolic derivatives

To build our inverse hyperbolic functions, we need to know how to find the inverse of a function in general, so let’s review.

To find the inverse of a function, we reverse the

$x$$y$

So for

$y=\cosh \left(x\right)$

We’d then solve this equation for $y$ by taking inverse hyperbolic cosine of both sides.

$x=\cosh \left(y\right)$

${\cosh }^{-1}x={\cosh }^{-1}\left[\cosh \left(y\right)\right]$

${\cosh }^{-1}x=y$

$y={\cosh }^{-1}x$

Remember that you can also see this function written as $y=\text{<mi mathvariant="normal" style="overflow-wrap: break-word;">a</mi><mi mathvariant="normal" style="overflow-wrap: break-word;">r</mi><mi mathvariant="normal" style="overflow-wrap: break-word;">c</mi><mi mathvariant="normal" style="overflow-wrap: break-word;">c</mi><mi mathvariant="normal" style="overflow-wrap: break-word;">o</mi><mi mathvariant="normal" style="overflow-wrap: break-word;">s</mi><mi mathvariant="normal" style="overflow-wrap: break-word;">h</mi>}\left(x\right)$. These are both representations of the inverse hyperbolic cosine function, and they can be used interchangeably.

Now that we understand how to find an inverse hyperbolic function when we start with a hyperbolic function, let’s talk about how to find the derivative of the inverse hyperbolic function.

Below is a chart which shows the six inverse hyperbolic functions and their derivatives.

$x=\cosh \left(y\right)$

$x=\cosh (y)$.