Examples of 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)

Differentiate both sides with respect to x, we have

\frac{{dy}}{{dx}} = \frac{d}{{dx}}{\cosh ^{ - 1}}\left( {{x^2} + 1} \right)

Using 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)

Differentiate 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}