## What is the Derivative of coth inverse x?

### Introduction

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 sin x and cos x. Or, we can directly find the derivative of arccot x by applying the first principle of differentiation. In this article, you will learn what the derivative of arccot x is and how to calculate the derivative of arccot x by using derivative rules.

## What is the derivative of arccoth x?

The derivative of arccot x with respect to the variable ‘x’ is equal to 11-x2. It is denoted by d/dx[arcoth x] or d/dx [coth-1x]. By definition, the hyperbolic function coth x consists of two exponential functions, ex and e-x such that:

$\coth x=\frac{1}{\tanh x}=\frac{e^x+e^{-x}}{e^x-e^{-x}}$

## Derivative of arccoth x formula

The formula of derivative of coth inverse x is equal to,

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

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

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

- Inverse function formula
- Implicit function theorem

## Derivative of arccoth using inverse function formula

Since the inverse function formula 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 arccoth by inverse function formula

To prove derivative of inverse sec, let us assume,

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

Then, we can write the above equation as;

$\coth y=x$

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

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

Where the derivative of coth x is -csch square. Now rearranging the above equation,

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

Since y = coth-1x.

$\frac{dy}{dx}=-\frac{1}{csch^2(\coth^{-1}x)}$

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

$csch^2y+1=csch^2y$

Or,

$csch^2\theta=1-\coth^2\theta=1-x^2$

Implies that

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

Substituting these values in the derivative formula,

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

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

## Derivative of coth inverse x by implicit function theorem

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

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

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

To prove the derivative of sec hyperbolic inverse function,

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

We can write it as,

$\coth y=x$

Or,

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

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

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

And,

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

By using implicit function theorem,

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

Since y = coth-1x.

$f'(x)=-\frac{1}{csch^2(\coth^{-1}x)}$

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

$\coth^2y+1=csch^2y$

Or,

$csch^2\theta=1-\coth^2\theta=1-x^2$

Implies that

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

Substituting these values in the derivative formula,

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

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

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

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

- Write the function as arccot x or coth-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 arccoth x.
- Now, select the variable by which you want to differentiate arccoth x. Here you have to choose ‘x’.
- Select how many times you want to differentiate the inverse coth x. In this step, you can choose 2 to find the second derivative, 3 to find the third derivative.
- Click on the calculate button. After this step, you will get the derivative of inverse coth x within a few seconds.