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