Derivative of Inverse Hyperbolic Tangent

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

Let the function of the form

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

By definition of 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}


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

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


From the above we have \tanh  y = x, by putting this 

\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}

Differentiation 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}

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