# Derivative of tanh inverse x

Learn about derivative of arctanh, formula, solution, how to calculate derivatives of arctanh with inverse and implicit functions theorem.

Alan Walker-

Published on 2023-05-26

## What is the Derivative of tanh inverse x?

Derivatives have a wide range of applications in almost every field of engineering and science. All derivatives of trigonometric functions can be found by following the derivative of sinh x and cosh x. Or, we can directly find the derivative of arctan x by applying the first principle of differentiation. In this article, you will learn what the derivative of arctan x is and how to calculate the derivative of arctan x by using different approaches.

## What is the derivative of arctanh x?

The derivative of tanh inverse x with respect to the variable ‘x’ is equal to 1/1-x^2. It is denoted by d/dx tanh-1x. By definition, the hyperbolic function tanh x consists of two exponential functions, ex and e-x such that:

$\tanh x=\frac{e^x-e^{-x}}{e^x+e^{-x}}$

## Derivative of arctan x formula

The formula of derivative of arctangent x is equal to,

$\frac{d}{dx}(\tanh^{-1})=\frac{1}{1-x^2}$

## How do you prove the derivative of tanh inverse x?

There are numerous ways to derive derivatives of tan x. Therefore, we can prove the derivative of arctan x by using;

1. Inverse function formula
2. Implicit function theorem

## Derivative of arctanh using inverse function formula

Since the formula of inverse function is a fundamental technique of finding derivatives of inverse functions. The inverse function formula to calculate derivative of a function f(x) can be written as:

$[f^{-1}](x)=\frac{1}{f'[f^{-1}(x)]}$

## Proof of derivative of arctanh by inverse function formula

To prove derivative of inverse sec, let us assume,

$y=\tan^{-1}x$

Then, we can write the above equation as;

$\tanh y=x$

Since, differentiating an equation of two independent variables is known as implicit differentiation, therefore from above equation,

$\DeclareMathOperator{\sech}{sech}\sech^2y\frac{dy}{dx}=1$

Where the derivative of tanh x is each square. Now rearranging the above equation,

$\frac{dy}{dx}=\frac{1}{\sech^2y}$

Since y = tanh-1x,

$\frac{dy}{dx}=\frac{1}{\sech^2(\tanh^{-1}x)}$

Assume that tanh-1x = θ then tanh θ = x, and since we know that,

$sech2\theta=1-\tanh^2\theta$

Or,

$\tanh^2\theta=1-\sech^2\theta=1-x^2$

Implies that

$\tanh^2\theta=1-x^2$

Substituting these values in the derivative formula,

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

Hence we have proved the derivative tanh inverse using the inverse function theorem. This theorem is suitable for differentiation of hyperbolic inverse function.

## Derivative of tanh inverse x by implicit function theorem

Since in implicit differentiation, we differentiate a function with two variables. Here we will prove the derivative of tanh inverse by using the implicit function theorem which is written as:

$f'(x)=-\frac{f_x}{f_y}$

### Proof of derivative of tanh-1(x) by implicit function theorem

To prove the derivative of sec hyperbolic inverse function,

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

We can write it as,

$\tanh y=x$

Or,

$f(x,y)=\tanh y-x$

Now we have to find the derivative of above expression with respect to x and y both,

$f_x=\frac{d}{dx}(\tanh y-x)=-1$

And,

$f_y=\frac{d}{dy}(\tanh y-x)=\sech^2y$

By using implicit function theorem,

$f'(x)=-\frac{f_x}{f_y}=\frac{1}{\sech^2y}$

Since y = tanh-1x.

$f'(x)=-\frac{1}{\sech^2(\tanh^{-1}x)}$

Assume that tanh-1x = θ then tanh θ = x, and since we know that,

$\sech^2\theta=1-\tanh^2\theta$

Or,

$\tanh^2\theta=1-\sech^2\theta=1-x^2$

Implies that

$\tanh^2\theta=1-x^2$

Substituting these values in the derivative formula,

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

Hence the derivative of arctanh x can be verified by using the theorem of implicit function.

## How to find the derivative of arctanh x with a calculator?

The easiest way to calculate the derivative of arctanh x is by using an online tool. You can use our derivative finder for this. Here, we provide you a step-by-step way to calculate derivatives by using the differential calculator.

1. Write the function as arctan x or tanh-1x in the “enter function” box. In this step, you need to provide input value as a function as you have to calculate the derivative of arctanh x.
2. Now, select the variable by which you want to differentiate arctanh x. Here you have to choose ‘x’.
3. Select how many times you want to differentiate inverse tangent x. In this step, you can choose 2 for second, 3 to find the third derivative.
4. Click on the calculate button. After this step, you will get the derivative of inverse tangent x within a few seconds.