# 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

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

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

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

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

Example: Find the derivative of

We have the given function as

Differentiating with respect to variable $x$, we get

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