# Examples of the Derivative of Inverse Hyperbolic Functions

Example: Differentiate ${\cosh ^{ – 1}}\left( {{x^2} + 1} \right)$ with respect to $x$.

Consider the function $y = {\cosh ^{ – 1}}\left( {{x^2} + 1} \right)$

Differentiating both sides with respect to $x$, we have
$\frac{{dy}}{{dx}} = \frac{d}{{dx}}{\cosh ^{ – 1}}\left( {{x^2} + 1} \right)$

Using the product rule of differentiation, we have
$\begin{gathered} \frac{{dy}}{{dx}} = \frac{1}{{\sqrt {{{\left( {{x^2} + 1} \right)}^2} – 1} }}\frac{d}{{dx}}\left( {{x^2} + 1} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{1}{{\sqrt {{x^4} + 2{x^2} + 1 – 1} }}\left( {2x} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{{2x}}{{\sqrt {{x^4} + 2{x^2}} }} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{{2x}}{{\sqrt {{x^2}\left( {{x^2} + 2} \right)} }} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{{2x}}{{x\sqrt {{x^2} + 2} }} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{2}{{\sqrt {{x^2} + 2} }} \\ \end{gathered}$

Example: Find $\frac{{dy}}{{dx}}$ if the given function is $y = {\sinh ^{ – 1}}\left( {\coth {x^2}} \right)$

We have the given function
$y = {\sinh ^{ – 1}}\left( {\coth {x^2}} \right)$

Differentiating both sides with respect to $x$, we have
$\begin{gathered} \frac{{dy}}{{dx}} = \frac{d}{{dx}}{\sinh ^{ – 1}}\left( {\coth {x^2}} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{1}{{\sqrt {1 + {{\left( {\coth {x^2}} \right)}^2}} }}\frac{d}{{dx}}\coth {x^2} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{1}{{\sqrt {1 + {{\coth }^2}{x^2}} }}\left( { – \csc {h^2}{x^2}} \right)\frac{d}{{dx}}{x^2} \\ \Rightarrow \frac{{dy}}{{dx}} = – \frac{{\csc {h^2}{x^2}}}{{\sqrt {1 + {{\coth }^2}{x^2}} }}\left( {2x} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = – \frac{{2x\csc {h^2}{x^2}}}{{\sqrt {1 + {{\coth }^2}{x^2}} }} \\ \end{gathered}$