## Introduction to the Derivative of csc inverse 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 csc x by applying the first principle of differentiation. In this article, you will learn what the derivative of csc inverse x is and how to calculate the derivative of cosecant inverse by using different approaches.

## What is the derivative of cosec inverse x?

**The derivative of inverse csc x, or the derivative of csc-1(x) is an important concept in calculus and trigonometry. It is expressed as, d/dx (csc-1x) = -1/|x|√(x^2-1).**

This formula represents the rate of change of the arccsc function, which is the inverse of the trigonometric function csc x. In a triangle, csc x is defined as the ratio of opposite to hypotenuse, and it can be expressed as 1/sin x which is written as;

$\csc x=\frac{1}{\sin x}$

Understanding the derivative of csc^-1(x) is important in fields such as calculus and physics.

## Derivative of csc-1x formula

The derivative of the csc^-1 is equal to -1/|x|√x^2-1. It can be expressed mathematically as:

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

In this formula, |x| is the absolute value of x which is used to avoid taking the derivative of undefined values.

## How do you prove the derivative of csc-1x?

There are multiple ways to derive the derivative csc inverse. We can prove the derivative of inverse csc x by using the following ways;

- First Principle
- Inverse function theorem

Each method provides a different way to compute the csc^-1 x derivative. By using these methods, we can mathematically prove the formula for finding the derivative of csc-1x.

## Derivative of csc inverse x by first principle

According to the first principle of derivative, the ln csc^-1 derivative is equal to -1/|x|√x^2-1. The derivative of a function by the first principle refers to finding a general expression for the curve line slope by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,

$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$

This formula allows us to determine the rate of change of a function at a specific point by using the limit definition of the derivative. You can also use our derivative by definition calculator as it also follows the above formula.

## Proof of derivative of csc inverse by first principle

To prove the derivative of csc-1x by using first principle, we start by replacing f(x) by csc-1x.

$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$

So,

$f'(x)=\lim_{h\to 0} \frac{\csc^{-1}(x+h)-\csc^{-1}(x)}{h}$

By using following trigonometric inverse formula,

$\sec^{-1}x+\csc^{-1}x=\frac{\pi}{2}$

We can rearrange it to get csc inverse,

$\csc^{-1}x=\frac{\pi}{2}-\sec^{-1}x$

Now using these substitution in the limit definition of derivative,

$f'(x)=\lim_{h\to 0}\frac{\frac{\pi}{2}-\sec^{-1}(x+h)-\left(\frac{\pi}{2}-\sec^{-1}x\right)}{h}$

Simplifying,

$f'(x)=\lim_{h\to 0}\frac{\frac{\pi}{2}-\sec^{-1}(x+h)-\frac{\pi}{2}+\sec^{-1}x}{h}=\lim_{h\to 0}\frac{-\sec^{-1}(x+h)+\sec^{-1}x}{h}$

Or,

$f'(x)=-\lim_{h\to 0}\frac{\sec^{-1}(x+h)-\sec^{-1}x}{h}$

Since the above expression is the rate of change of secant inverse, therefore, using the value of the sec inverse derivative,

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

## Derivative of csc inverse using inverse function formula

The inverse function formula is a fundamental technique for finding the derivatives of inverse functions. Mathematically this formula is expressed as:

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

This formula is useful to calculate the rate of change of inverse trigonometric functions, such as csc^-1(x). By using the inverse function formula, we can derive the derivative of csc^-1(x) in terms of the derivative of csc x.

## Proof of derivative of arccsc by inverse function formula

To prove derivative of inverse csc, we start by assuming that,

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

Then, we can write the above equation as;

$\csc y=x$

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

$-\csc y\cot y\frac{dy}{dx}=1$

Where the derivative of csc x is -csc xcot x. Now rearranging the above equation,

$\frac{dy}{dx}=-\frac{1}{\csc y\cot y}$

Since y = csc-1x.

$\frac{dy}{dx}=-\frac{1}{\csc(\csc^{-1}x)\cot(\csc^{-1}x)}$

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

$\csc^2\theta=\cot^2\theta+1$

Or,

$\cot^2\theta=\csc^2\theta-1=x^2-1$

Taking Square root,

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

Similarly,

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

Substituting these values in the derivative formula,

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

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

## How to find the derivative of arccsc(x) with a calculator?

The easiest way to calculate the derivative of cosecant 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.

- Write the function as csc-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 csc-1x.
- Now, select the variable by which you want to differentiate csc-1x. Here you have to choose ‘x’.
- Select how many times you want to differentiate cosecant inverse x. In this step, you can choose 2 for the second derivative, 3 for the third derivative and so on.
- Click on the calculate button.

After this step, the derivative of inverse function calculator will provide you with the derivative of csc inverse x within a few seconds.