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}