Question

To find the derivative of the function.

Short answer

The derivative of the function \(y = {\cosh ^{ - 1}}\sqrt x \) is, \(\frac{{dy}}{{dx}} = \frac{1}{{2\sqrt x \sqrt {x - 1} }}\).

Step-by-step solution

 Given function

The function is \(y = {\cosh ^{ - 1}}\sqrt x \).

The Concept of derivative and hyperbolic function

Derivative rules:

(1) The chain rule:\(\frac{{dy}}{{dx}} = \frac{{dy}}{{du}}\frac{{du}}{{dx}}\)

(2) The derivative of the inverse hyperbolic cosine function is,\(\frac{d}{{dx}}\left( {cos{h^{ - 1}}x} \right) = \frac{1}{{\sqrt {{x^2} - 1} }}\).

Evaluate the derivative by the use of concepts

The derivative of the function \(y = {\cosh ^{ - 1}}\sqrt x \) is computed as shown below.

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

Let, \(u = \sqrt x \).

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

Apply the chain rule and simplify the terms.

\(\begin{array}{c}\frac{{dy}}{{dx}} = \frac{d}{{du}}\left( {\cos {h^{ - 1}}u} \right)\frac{{du}}{{dx}}\\ = \frac{1}{{\sqrt {{u^2} - 1} }}\frac{{du}}{{dx}}\\{\rm{Q}}\frac{d}{{dx}}\left( {\cos {h^{ - 1}}x} \right) = \frac{1}{{\sqrt {{x^2} - 1} }}\end{array}\)

Substitute the value, \(u = \sqrt x \).

\(\begin{array}{c}\frac{{dy}}{{dx}} = \frac{1}{{\sqrt {{{(\sqrt x )}^2} - 1} }}\frac{d}{{dx}}(\sqrt x )\\ = \frac{1}{{\sqrt {x - 1} }}\left( {\frac{1}{{2\sqrt x }}} \right)\\ = \frac{1}{{2\sqrt x \sqrt {x - 1} }}\\ = \frac{1}{{2\sqrt {x(x - 1)} }}\end{array}\)

Therefore, the derivative of the function \(y = {\cosh ^{ - 1}}\sqrt x \) is, \(\frac{{dy}}{{dx}} = \frac{1}{{2\sqrt {x(x - 1)} }}\).