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;

1. Inverse function formula
2. 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.

1. 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.
2. Now, select the variable by which you want to differentiate arccoth x. Here you have to choose ‘x’.
3. 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.
4. Click on the calculate button. After this step, you will get the derivative of inverse coth x within a few seconds.