Derivative of Inverse Hyperbolic Cosine

In this tutorial we shall be concerned with the derivative of inverse hyperbolic cosine function with an example.

Let the function of the form

y = f\left( x \right) = {\cosh ^{ - 1}}x

By definition of 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}


We can write from the fundamental rules of inverse hyperbolic identities \sinh y = \sqrt {{{\cosh }^2}y - 1} . Putting this value in 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, by putting this 

\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

Differentiation 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}