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;

1. First Principle
2. 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.

1. 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.
2. Now, select the variable by which you want to differentiate csc-1x. Here you have to choose ‘x’.
3. 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.
4. 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.