Derivative of csc square x

Learn what is the derivative of csc square x with formula. Also understand how to prove the derivative of csc^2x by different techniques.

Alan Walker-

Published on 2023-05-26

Introduction to the Derivative of csc squared x

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of csc^2(x) can be calculated by following the rules of differentiation. Or, we can directly find the d/dx csc^2x by applying the first principle of differentiation. In this article, you will learn what the derivative of csc squared is and how to calculate the csc^2x derivative by using different approaches.

What is the derivative of csc^2x?

The derivative of csc^2(x) with respect to x is -2csc^2(x)cot(x). This is denoted by d/dx(csc^2(x)) and represents the rate of change of the trigonometric function cosecant. Cosecant is defined as the ratio of the hypotenuse to the opposite side in a triangle, and can be written as;

csc(x) = 1/sin(x)

Knowing how to calculate the csc2x derivative is essential in calculus and other areas of mathematics. Different derivative rules can be used to find the csc squared derivative.

Derivative of csc2 x formula

The derivative of csc squared x can be calculated using the formula;

d/dx(csc^2(x)) = -2csc^2(x)cot(x)

This formula shows that the derivative of cosecant squared x is equal to the product of the cosecant squared and cotangent functions. It can also be expressed as the negative of 2 times csc^2(x)cot(x).

How do you prove the derivative of csc^2 x?

There are different ways to derive derivatives of csc2 x. Therefore, we can prove the derivative of csc2 x by using;

  1. Chain Rule
  2. Product rule

Each method provides a different way to compute the csc squared derivative. By using these methods, we can mathematically prove the formula for finding csc2x derivative.

Derivative of csc^2x by chain rule

To calculate the csc squared x differentiation, we can use the chain rule since the cosine function can be expressed as a combination of two functions. The chain rule of derivatives states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. The chain rule of derivatives is defined as;

dy / dx = dy / du x du / dx

You can also use a differential dy calculator to differentiate csc^2x wuth just one click.

Proof of csc^2x derivative by chain rule

To prove the d/dx csc^2x by using chain rule, we start by assuming that,

y = u2 where u = csc x

By chain rule,

y= 2u. du/dx


du/dx = -2u

now, using the value of u and du/dx in y, we have,

y= - 2csc2 x.cot x

The csc2x derivative can also be verified by using chain rule.

Derivative of csc2 x by using product rule

The product rule is a formula used in calculus to calculate the derivative of the product of two functions. Specifically, the product rule is used when you need to differentiate two functions that are multiplied together. The formula for the product rule is:

d/dx(uv) = u(dv/dx) + (du/dx)v

In this formula, u and v are functions of x, and du/dx and dv/dx are their respective derivatives with respect to x.

Proof of derivative of csc^2(x) by using product rule

To prove the derivative of csc squared x by using product rule of derivatives, we have,

d/dx (csx) = d/dx(csc x)

d/dx (csc2x) = csc x. (csc x)+ csc x. (csc x)

d/dx (csc2x) = csc x.(- csc x.cot x)+ csc x.(- csc x.cot x)

It can be written as;

d/dx (csc2x) = - 2csc2 x.cot x

Hence the d/dx csc^2x is proved by using the product rule.

How to find the derivative of csc^2x with a calculator?

The easiest way to calculate the derivative of csc 2 is by using an online tool. You can use our derivative calculator with steps for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

  1. Write the function as csc^2x in the enter function box. In this step, you need to provide input value as a function as you have to calculate the csc2x derivative.
  2. Now, select the variable by which you want to differentiate csc^2x. Here you have to choose x.
  3. Select how many times you want to differentiate csc^2x. In this step, you can choose 2 for second, 3 for 3rd order derivative and so on.
  4. Click on the calculate button. After this step, you will get the derivative of csc^2x within a few seconds.

After completing these steps, you will receive the derivative of csc squared within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.

Frequently asked questions

How do you find the derivative of sec2x?

The sec2x differentiation can be calculated as;

d/dx (sec2x) = 2 sec 2x.tan 2x

What is the derivative of csc inverse?

The inverse cosecant function's derivative with respect to is equal to the negative reciprocal of the product of the modulus of and the square root of the subtraction of one from squared.

What is the second derivative of cos?

Differentiating the first derivative of cos x, -sin x, gives the second derivative of cos x. -cos x is the derivative of -sin x. As a result, cos x's second derivative is -cos x. Also you can use 2nd derivative calculator to find derivatives easily.

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