# Derivative of cot square x

Learn how to calculate the derivative of cot^2x with easy steps. Also verify the derivative of cot square x by using chain rule and product rule.

Alan Walker-

Published on 2023-05-26

## Introduction to the Derivative of cot square x

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

## What is the derivative of cot^2x?

The derivative of cot^2x with respect to the variable x is equal to the square of -cosec x. It is denoted by d/dx(cot2 x). It is the rate of change of the trigonometric function cot^2x.

In a triangle, it is the ratio of cosine and sine function. It is written as;

Cot^2x =cos2 x/ sin2 x

## Derivative of cot^2x formula

The formula of derivative of cot^2x is equal to the negative of the cosecant function, that is;

d/dx (cot2 x) = -2cot x cosec2 x

## How do you prove the derivative of cot^2x?

There are numerous ways to derive derivatives of cot^2x. Therefore, we can prove the derivative of cot^2x by using;

1. Chain Rule
2. Quotient Rule

## Derivative of cot^2x by chain rule

The derivative of cot^2x can be calculated by using chain rule because the cosine function can be written as the combination of two functions. The chain rule of derivative is defined as;

dy/dx = dy/du x du/dx

## Proof of derivative of cot^2x by chain rule

To prove the derivative of cot^2x by using chain rule, assume that,

f(x) = cot2 x = 1/tan2 x = (tan x)-2

By using chain rule of differentiation,

f(x) = (-2) (tan x)-2 d/dx (sec2 x)

Simplifying,

f(x) = (-2/tan2 x) . (sec2 x)

Again,

f(x) = -2sec2 x/ tan2 x

Since sin x / cos x =tan x and 1/cos x = sec x, therefore we have

f(x) = -2cot x cosec2 x

## Derivative of cot^2x using quotient rule

Since the cotangent is the reciprocal of the tangent. Therefore, the derivative of cot can also be calculated by using the quotient rule. The quotient rule is defined as;

d/dx (f/g) = f(x). g(x) -g(x).f(x) /{g(x)}2

## Proof of derivative of cot^2x by quotient rule

To prove the derivative of cot^2x, we can write it,

f(x) = cot2 x=1/ tan2 x =u/v

Supposing that u = 1 and v = tan2 x. Now by quotient rule,

f(x) = (vu - uv)/v2

f'(x) = [tan x d/dx(1) - 1. d/dx(tan x)] / (tan2 x)2

= [tan2 x (0) - 1 (2tan xsec2 x)] / tan4x

= (-2tan xsec2 x ) / tan4x

= -2cot x. cosec2 x

Hence, we have derived the derivative of cot^2x using the quotient rule calculator.

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

The easiest way to calculate the derivative of cot^2x 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 cot^2x 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 cot^2x.
2. Now, select the variable by which you want to differentiate cot^2x. Here you have to choose x.
3. Select how many times you want to differentiate cotangent 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, you will get the derivative of c x within a few seconds.

### What are derivative formulas?

Power Rule: (d/dx) (xn ) = nxn-1

Derivative of a constant, a: (d/dx) (a) = 0.

Derivative of a constant multiplied with function f: (d/dx) (a. f) = af'

Sum Rule: (d/dx) (f ± g) = f' ± g'

Product Rule: (d/dx) (fg)= fg' + gf'

Quotient Rule: d d x ( f/g ) = g f ' - f g ' /g2 .

### How do you differentiate cot squared?

The derivative of cot squared can be calculated by using power rule such as;

d/dx(cot2 x) = - 2cot x. cosec2x

cot squared x can also be differentiated by using the product rule of derivatives.