Derivative of cot x

Learn how to calculate the derivative of a cot x by first principle with easy steps. Also verify the derivative of cotx by using chain rule and product rule.

Alan Walker-

Published on 2023-05-26

Introduction to the Derivative of cot x

Derivatives have a wide range of applications in almost every field of engineering and science. The cotx differentiation can be calculated by following the rules of differentiation.

Or, we can directly find the derivative of cotx by applying the first principle of differentiation. In this article, you will learn what the cot x derivative is and how to calculate the derivative of cotx by using different approaches.

What is the derivative of cotx?

The derivative of cot x with respect to the variable x is denoted by d/dx(cot x) and is equal to the negative square of cosec x. This derivative represents the rate of change of the trigonometric function cot x. In a triangle, it is the ratio of cosine and sine functions. To find the cotx derivative, we can use the formula;
cot x = cos x / sin x
Understanding the differentiation of cot x is essential in calculus and trigonometry, and it can help in solving various mathematical problems involving trigonometric functions.

Derivative of cot x formula

The formula for the d/dx cot x is equal to the negative of the square of the cosecant function, which can be expressed as:

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

Understanding this formula is important in calculus and trigonometry, as it helps in finding the rate of change of cot x with respect to x. By using this formula and applying the chain rule of differentiation, we can also find the derivative of more complex functions involving cot x. Remembering this formula and its application can make solving mathematical problems involving cot x and related trigonometric functions more manageable.

How do you prove the derivative of cotx?

There are various ways to prove the cot x derivative. These are;

  1. First Principle

  2. Chain Rule

  3. Quotient Rule

Each method provids a different way to compute the differentiation of cotx. By using these methods, we can mathematically prove the formula for finding differential of cot x.

Cotx differentiation by first principle

A fundamental way to find the derivative of a function is by using the first principle, which is also known as the delta method. This method involves finding a general expression for the slope of a curve by using algebra. The derivative is a measure of the instantaneous rate of change of a function at a specific point, and it can be calculated using the limit formula:

f(x) = lim f(x + h) - f(x) / h

This formula represents the slope of the tangent line to the curve of the function at the point x. Understanding the first principle can be helpful in finding the derivative of cotx or other functions, especially when other methods like the chain rule or quotient rule are not applicable.

Proof of cotx derivative by first principle

To prove the derivative of cot x by using first principle, we start by replacing f(x) by cot x. f(x) = limh→0f(x + h) - f(x) / h

f(x) = lim cot (x + h) - cot x / h

Similarly, you can replace f(x) by cot 2x to calculate derivative of cot 2x.

Since cot x = cos x / sin x, therefore,

f(x) = lim cos(x + h) /sin (x + h) - cos x / sin x / h

More simplification,

f(x) = lim sin x cos(x + h) - sin (x + h) cos x / hsin x.sin (x + h)

Now, by the trigonometric formula, sin A cos B - cos A sin B = sin (A - B)

f(x) = lim sin (x - (x + h)) / hsin x.sin (x + h)

f(x) = lim sin ( - h) / hsin x.sin (x + h)

We have sin( - h) = - sin h, therefore,

f(x) = - lim sin h/ h * lim 1/[sin x. sin (x + h)]

When h approaches to zero,

f(x) = - 1 * 1/sin x. sin x

We can write it as;

f(x) = - cosec2x

This is how we can proof cotx differentiation by using the first principle. Also, we can use the derivative definition calculator to prove the derivative of cot(x). 

Differentiation of cot x by chain rule

To calculate the cot x differentation, 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

Where, u=g(x), y=f(u), dy/du is the derivative of f(u) with respect to u and du/dx is the derivative of g(x) with respect to x.

Derivative of cotx proof by chain rule

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

f(x) = cot x = 1/tan x = (tan x) - 1

By using chain rule of differentiation,

f(x) = ( - 1) (tan x) - 2 d / dx (sec2 x)


f(x) = ( - 1/tan2x) . (sec2 x)


f(x) = - sec2 x / tan2x

Finally, we can use the trigonometric identity cot x = cos x / sin x and simplify further to get:

f(x) = - cosec2 x

Use the chain rule finder to calculate the cot derivative easily online. 

Derivative of cot x using quotient rule

Another method for finding the derivative of csc x is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. Since the secant function is the reciprocal of cosine, the derivative of cosecant can also be calculated 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 differentiation of cot x by quotient rule

To prove the derivative of cotx, we can start by writing it as,

f(x) = cot x = 1/ tan x = u/v

Supposing that u = 1 and v = tan x. Now we can then apply the quotient rule of differentiation, which states that the derivative of a quotient of two functions is equal to:

f(x) = (vu - uv) / v^2

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

f'(x)= [tan x (0) - 1 (sec2 x)] / tan2x

f'(x)= ( - sec2 x ) / tan2x

f'(x)= - cosec2 x

Hence, we have derived the cot derivative using the quotient rule of differentiation calculator.

How to find the differentiation of cotx with a calculator?

The easiest way to calculate the differentiation of cot x 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 x in the enter function box. In this step, you need to provide input value as a function as you have to calculate the cot x derivative.

  2. Now, select the variable by which you want to differentiate cot x. 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.

After completing these steps, you will receive the cotx derivative within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions. Also, the derivative at a point calculator allows you to find the rate of change of a function at a specific point. 

Frequently Asked Questions

What is derivative of cos square x by first principle?

The derivative from first principle is the measure of rate of change in a function. The derivative of cos square x is equal to the negative of 2 cos x.sin x and represented as;

d / dx (cos2x) = - 2cos x. sin x

What is cot x equal to?

The cot x is equal to the reciprocal of tangent function. Since the tangent is equal to the ratio of sine and cosine. Therefore, cotangent is written as;

Cot x = cos x / sin x

What is the second derivative of cot x?

The first derivative of cotangent is equal to the negative of the square of cosec x. And the second derivative of cot can be calculated by differentiating first derivative. Therefore, the second derivative of cot(x) is;

d2 / dx2 (cot x) = 2cot x cosec2 x;

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