## 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-1x

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.