# Derivative Rule of Inverse Hyperbolic Tangent function

## Formula

$\dfrac{d}{dx}{\,\tanh^{-1}{x}}$ $\,=\,$ $\dfrac{1}{1-x^2}$

### Introduction

The inverse hyperbolic tangent is written in function form as $\tanh^{-1}{(x)}$ or $\operatorname{arctanh}{(x)}$ if the literal $x$ represents a variable. The differentiation of the inverse hyperbolic tan function with respect to $x$ is written in the following mathematical forms.

$(1).\,\,\,$ $\dfrac{d}{dx}{\, (\tanh^{-1}{x})}$

$(2).\,\,\,$ $\dfrac{d}{dx}{\, (\operatorname{arctanh}{x})}$

In simple form, the derivative of inverse hyperbolic tan function is written as $(\tanh^{-1}{x})’$ or $(\operatorname{arctanh}{x})’$ mathematically in differential calculus.

The differentiation of hyperbolic inverse tangent function with respect to $x$ is equal to multiplicative inverse of difference of $x$ squared from one.

$\implies$ $\dfrac{d}{dx}{\, \tanh^{-1}{x}}$ $\,=\,$ $\dfrac{1}{1-x^2}$

#### Other forms

The derivative of inverse hyperbolic tangent function can also be expressed in any variable in mathematics.

#### Example

$(1) \,\,\,$ $\dfrac{d}{dl}{\, \tanh^{-1}{l}}$ $\,=\,$ $\dfrac{1}{1-l^2}$

$(2) \,\,\,$ $\dfrac{d}{dq}{\, \tanh^{-1}{q}}$ $\,=\,$ $\dfrac{1}{1-q^2}$

$(3) \,\,\,$ $\dfrac{d}{dy}{\, \tanh^{-1}{y}}$ $\,=\,$ $\dfrac{1}{1-y^2}$

### Proof

Learn how to prove differentiation rule of inverse hyperbolic tangent function from the first principle of differentiation.

Latest Math Topics
Jun 26, 2023
Jun 23, 2023

Latest Math Problems
Jul 01, 2023
Jun 25, 2023
###### Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Practice now

###### Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

###### Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.