Derivative of Inverse Hyperbolic Secant

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

Let the function of the form

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

By definition of inverse trigonometric function, y = {\operatorname{sech} ^{ - 1}}x can be written as

\operatorname{sech} y = x

Differentiating both sides with respect to the variable x, we have

\begin{gathered} \frac{d}{{dx}}\operatorname{sech} y = \frac{d}{{dx}}\left( x \right) \\ \Rightarrow - \sec {\text{h}}y\tanh y\frac{{dy}}{{dx}} = 1 \\ \Rightarrow \frac{{dy}}{{dx}} = - \frac{1}{{\sec {\text{h}}y\tanh y}}\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right) \\ \end{gathered}

We can write from the fundamental rules of inverse hyperbolic identities \tanh y = \sqrt {1 - \sec {{\text{h}}^2}x} . Putting this value in above relation (i) and simplifying, we have

\frac{{dy}}{{dx}} = - \frac{1}{{\sec {\text{h}}x\sqrt {1 - \sec {{\text{h}}^2}x} }}

From the above we have \operatorname{sech} y = x, by putting this

\begin{gathered} \frac{{dy}}{{dx}} = - \frac{1}{{x\sqrt {1 - {x^2}} }} \\ \Rightarrow \frac{d}{{dx}}\left( {{{\operatorname{sech} }^{ - 1}}x} \right) = - \frac{1}{{x\sqrt {1 - {x^2}} }} \\ \end{gathered}

Example: Find the derivative of

y = f\left( x \right) = {\operatorname{sech} ^{ - 1}}\sqrt x

We have the given function as

y = {\operatorname{sech} ^{ - 1}}\sqrt x

Differentiation with respect to variable x, we get

\frac{{dy}}{{dx}} = \frac{d}{{dx}}{\operatorname{sech} ^{ - 1}}\sqrt x

Using the rule, \frac{d}{{dx}}\left( {{{\operatorname{sech} }^{ - 1}}x} \right) = - \frac{1}{{x\sqrt {1 - {x^2}} }}, we get

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