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
\[\begin{gathered} \frac{{dy}}{{dx}} = \frac{d}{{dx}}{\sinh ^{ – 1}}\left( {\coth {x^2}} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{1}{{\sqrt {1 + {{\left( {\coth {x^2}} \right)}^2}} }}\frac{d}{{dx}}\coth {x^2} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{1}{{\sqrt {1 + {{\coth }^2}{x^2}} }}\left( { – \csc {h^2}{x^2}} \right)\frac{d}{{dx}}{x^2} \\ \Rightarrow \frac{{dy}}{{dx}} = – \frac{{\csc {h^2}{x^2}}}{{\sqrt {1 + {{\coth }^2}{x^2}} }}\left( {2x} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = – \frac{{2x\csc {h^2}{x^2}}}{{\sqrt {1 + {{\coth }^2}{x^2}} }} \\ \end{gathered} \]