# Derivative of Inverse Hyperbolic Cosine

In this tutorial we shall discuss the derivative of the inverse hyperbolic cosine function with an example.

Let the function be of the form $y = f\left( x \right) = {\cosh ^{ – 1}}x$

By the definition of the inverse trigonometric function, $y = {\cosh ^{ – 1}}x$ can be written as
$\cosh y = x$

Differentiating both sides with respect to the variable $x$, we have
$\begin{gathered} \frac{d}{{dx}}\cosh y = \frac{d}{{dx}}\left( x \right) \\ \Rightarrow \sinh y\frac{{dy}}{{dx}} = 1 \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{1}{{\sinh y}}\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered}$

From the fundamental rules of inverse hyperbolic identities, this can be written as $\sinh y = \sqrt {{{\cosh }^2}y – 1}$. Putting this value in the above relation (i) and simplifying, we have
$\frac{{dy}}{{dx}} = \frac{1}{{\sqrt {{{\cosh }^2}y – 1} }}$

From the above, we have $\cosh y = x$, thus
$\begin{gathered} \frac{{dy}}{{dx}} = \frac{1}{{\sqrt {{x^2} – 1} }} \\ \Rightarrow \frac{d}{{dx}}\left( {{{\cosh }^{ – 1}}x} \right) = \frac{1}{{\sqrt {{x^2} – 1} }} \\ \end{gathered}$

Example: Find the derivative of $y = f\left( x \right) = {\cosh ^{ – 1}}\sqrt x$

We have the given function as
$y = {\cosh ^{ – 1}}\sqrt x$

Differentiating with respect to variable $x$, we get
$\frac{{dy}}{{dx}} = \frac{d}{{dx}}{\cosh ^{ – 1}}\sqrt x$

Using the rule, $\frac{d}{{dx}}\left( {{{\cosh }^{ – 1}}x} \right) = \frac{1}{{\sqrt {{x^2} – 1} }}$, we get
$\begin{gathered} \frac{{dy}}{{dx}} = \frac{1}{{\sqrt {{{\left( {\sqrt x } \right)}^2} – 1} }}\frac{d}{{dx}}\sqrt x \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{1}{{\sqrt {x – 1} }}\frac{1}{{\sqrt x }} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{1}{{\sqrt {{x^2} – x} }} \\ \end{gathered}$