What is the Derivative of arcsinh x?

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

What is the derivative of sinh-1 x?

The derivative of sinh inverse with respect to the variable ‘x’ is equal to cosh x. It is denoted by d/dx (sin-1x). It is the inverse of the rate of change of the hyperbolic inverse function sinh x.

By definition, the hyperbolic function sinh x consists of two exponential functions, e^x and e^-x such that:

$\sinh x=\frac{e^x-e^{-x}}{2}$

Derivative of sinh-1 x formula

The derivative formula of sinh inverse x is equal to the negative of the derivative of cos inverse, that is;

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

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

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

1. First Principle
2. Implicit differentiation

Derivative of sin 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:

$[f^{-1}]'(x)=\frac1f'[f-1(x)]$

Proof of derivative of arcsin by inverse function formula

To prove derivative of inverse sinh, let us assume,

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

Then, we can write the above equation as;

$\sinh y=x$

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

$\cosh y\frac{dy}{dx}=1$

Where the derivative of sinh x is cosh x. Now rearranging the above equation,

$\frac{dy}{dx}=\frac{1}{\cosh y}$

Since y = sinh-1x. 

$\frac{dy}{dx}=\frac{1}{\cosh(\sinh^{-1}x)}$

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

$\cosh^2\theta=1+\sinh^2\theta=1+x^2$

Taking Square root, 

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

Substituting these values in the derivative formula, 

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

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

Derivative of sinh inverse x by implicit function theorem

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

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

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

To prove the derivative of sin hyperbolic inverse function, 

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

We can write it as,

$\sinh y=x$

Or, 

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

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

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

And, 

$f_y=\frac{d}{dy}(\sinh y-x)=\cosh y$

By using implicit function theorem, 

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

Since y = sinh-1x. 

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

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

$\cosh^2\theta=1+\sinh^2\theta=1+x^2$

Taking Square root, 

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

Substituting these values in the derivative formula, 

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

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

How to find the derivative of arcsinh x with a calculator?

The easiest way to calculate the derivative of sinh 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 sin-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 sinh-1x.
2. Now, select the variable by which you want to differentiate sinh-1x. Here you have to choose ‘x’.
3. Select how many times you want to differentiate sine hyperbolic inverse x. In this step, you can choose 2 for second, 3 to find the third derivative. 
4. Click on the calculate button. After this step, you will get the derivative of sine inverse x within a few seconds.