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