Derivative of Inverse Hyperbolic Tangent

In this tutorial we shall discuss the derivative of the inverse hyperbolic tangent function with an example.

Let the function be of the form

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

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

\tanh y = x

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

\begin{gathered} \frac{d}{{dx}}\tanh y = \frac{d}{{dx}}\left( x \right) \\ \Rightarrow {\operatorname{sech} ^2}y\frac{{dy}}{{dx}} = 1 \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{1}{{{{\operatorname{sech} }^2}y}}\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right) \\ \end{gathered}

From the fundamental rules of inverse hyperbolic identities, this can be written as  {\operatorname{sech} ^2}y = 1 - {\tanh ^2}y. Putting this value in the above relation (i) and simplifying, we have

\frac{{dy}}{{dx}} = \frac{1}{{1 - {{\tanh }^2}y}}

 

From the above we have \tanh y = x, thus

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

 

Example: Find the derivative of

y = f\left( x \right) = {\tanh ^{ - 1}}{x^2}

We have the given function as

y = {\tanh ^{ - 1}}{x^2}

Differentiating with respect to variable x, we get

\frac{{dy}}{{dx}} = \frac{d}{{dx}}{\tanh ^{ - 1}}{x^2}

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

\begin{gathered} \frac{{dy}}{{dx}} = \frac{1}{{1 - {{\left( {{x^2}} \right)}^2}}}\frac{d}{{dx}}{x^2} \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{1}{{1 - {x^4}}}2x \\ \Rightarrow \frac{{dy}}{{dx}} = \frac{{2x}}{{1 - {x^4}}} \\ \end{gathered}