What is the Derivative of csch inverse x?

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of hyperbolic csc inverse x can be calculated by following the derivative rules. Or, we can directly find the derivative of inverse csc x by applying the first principle of differentiation. In this article, you will learn what the derivative of csch inverse x is and how to calculate the derivative of cosecant inverse by using different approaches. 

What is the derivative of csch-1 x?

The derivative of csch^-1(x) with respect to the variable ‘x’ is equal to -1/|x|√x^2+1. It is denoted by d/dx (csch-1x). It is the inverse of the rate of change of the inverse hyperbolic function csch x. By definition, the hyperbolic function csch x consists of two exponential functions, e^x and e^-x such that:

$\DeclareMathOperator{\csch}{csch}\csch x=\frac{1}{\sinh x}=\frac{2}{e^x+e^{-x}}$

Derivative of csch-1 x formula

The formula of derivative of csc inverse x is equal to the -1/|x|√x^2-1, that is;

$\frac{d}{dx}[\DeclareMathOperator{\csch}{csch}\csch^{-1}x]=-\frac{1}{|x|\sqrt{x^2+1}}$

How do you prove the derivative of csch-1x?

 There are numerous ways to derive derivatives of csch inverse x. Therefore, we can prove the derivative of csch^-1(x) by using;

1. Inverse function formula 
2. Implicit function theorem

Derivative of csch inverse using inverse function formula

Since the inverse function derivative 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 arc csch by inverse function formula

To prove derivative of inverse sec, let us assume,

$y = \DeclareMathOperator{\csch}{csch}\csch^{-1}x$

Then, we can write the above equation as;

$\csch y=x$

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

$-\DeclareMathOperator{\csch}{csch}\csch y \coth y\frac{dy}{dx}=1$

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

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

Since y = csch-1x. 

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

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

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

Or, 

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

Taking Square root, 

$\coth \theta=\sqrt{1+x^2}$

Similarly, 

$\csch(\csch^{-1}x)=\csch\theta =x$

Substituting these values in the derivative formula, 

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

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

Derivative of csch inverse x by implicit function theorem

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

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

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

To prove the derivative of sec hyperbolic inverse function, 

$y=\DeclareMathOperator{\csch}{csch}\csch^{-1}x$

We can write it as,

$\csch y=x$

Or, 

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

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

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

And, 

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

By using implicit function theorem, 

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

Since y = csch-1x. 

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

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

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

Or, 

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

Taking Square root, 

$\coth \theta=\sqrt{1+x^2}$

Similarly, 

$\csch(\csch^{-1}x)=\csch\theta=x$

Substituting these values in the derivative formula, 

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

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

How to find the derivative of arc csch x with a calculator?

The easiest way to calculate the derivative of hyperbolic secant inverse 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 csch-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 csch-1x.
2. Now, select the variable by which you want to differentiate csch-1x. Here you have to choose ‘x’.
3. Select how many times you want to differentiate csch inverse x. In this step, you can choose 2 for second, 3 for third derivative and so on.

Click on the calculate button. After this step, the inverse function derivative calculator will provide you the derivative of csch inverse x within a few seconds.