What is the Derivative of arcsech x?

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

What is the derivative of sech-1 x?

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

$\DeclareMathOperator{\sech}{sech}\sech x=\frac{1}{\cosh x}=\frac{2}{e^x+e^{-x}}$

Derivative of sech-1 x formula

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

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

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

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

1. Inverse function formula 
2. Implicit function theorem

Derivative of sech inverse 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 arcsec by inverse function formula

To prove derivative of inverse sec, let us assume,

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

Then, we can write the above equation as;

$\sech y=x$

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

$-\tanh y\sech y\frac{dy}{dx}=1$

Where the derivative of sech x is -sech xtanh x. Now rearranging the above equation,

$\frac{dy}{dx}=-\frac{1}{\tanh y\sech y}$

Since y = sech-1x. 

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

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

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

Or, 

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

Taking Square root, 

$\tanh\theta=\sqrt{1-x^2}$

Similarly, 

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

Substituting these values in the derivative formula, 

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

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

Derivative of sech inverse x by implicit function theorem

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

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

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

To prove the derivative of sec hyperbolic inverse function, 

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

We can write it as,

$\sech y=x$

Or, 

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

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

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

And, 

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

By using implicit function theorem, 

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

Since y = sech-1x. 

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

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

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

Or, 

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

Taking Square root, 

$\tanh\theta=\sqrt{1-x^2}$

Similarly, 

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

Substituting these values in the derivative formula, we get

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

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

How to find the derivative of arcsech 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 differentiation calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

1. Write the function as sech-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 sech-1x.
2. Now, select the variable by which you want to differentiate sech-1x. Here you have to choose ‘x’.
3. Select how many times you want to differentiate sech inverse x. In this step, you can choose 2 for second, 3 for third derivative and so on.
4. Click on the calculate button. After this step, the derivative of inverse function calculator will provide you the derivative of sech inverse x within a few seconds.