Derivative of Cot x

Derivative of Cot x is -cosec²x. It refers to the process of finding the change in the sine function with respect to the independent variable. Derivative of cot x is also known as differentiation of cot x which is the process of finding rate of change in the cot trigonometric function.

In this article, we will learn about the derivative of cot x and its formula including the proof of the formula using the first principle of derivatives, quotient rule, and chain rule as well.
👁 Derivative-of-Cot-x

What is Derivative of Cot x?

The derivative of cot x is -cosec²x. The derivative of cot x is one of the six trigonometric derivatives that we have to study. It is the differentiation of trigonometric function cotangent with respect to the variable x in the present case. If we have cot y or cot θ then we differentiate the cotangent with respect to y or θ respectively.

Learn More:

• Calculus in Maths
• Derivative in Math

Derivative of Cot x Formula

The formula of the derivative of cot x is given by:

(d/dx)[cot x] = -cosec²x

or

(cot x)' = -cosec²x

Proof of Derivative of Cot x

The derivative of cot x can be proved using the following ways:

• By using First Principle of Derivative
• By using Quotient Rule
• By using Chain Rule

Derivative of Cot x by First Principle of Derivative

Let's start the proof for derivative of Cot x:

Let f(x) = Cot x

By the First Principle of Derivative

f'(x)= lim h→0 f(x+h)-f(x)/h

= lim h→0 cot(x+ h)- cot x/ h

= lim h→0 [cos(x+h)/sin(x+h)- cos x/ sin x]/h

= lim h→0 sin x cos(x+h)-cos x sin (x+h) / sin(x+h) sin x. h

=lim h→0 sin [x-(x+h) / sin(x+h).sin x .h

= lim h→0 - sin h/h lim h→0 1/sin (x+h)sin x

= -1 × 1/sinx. sinx

= -1/ sin²x

= -cosec²x

Derivative of Cot x by Quotient Rule

To find the derivative of cot x using the quotient rule of derivative we have to use the following mentioned formulas

• (d/dx) [u/v] = [u’v – uv’]/v²
• sin²(x)+ cos²(x)= 1
• cot x = cos x / sin x
• cosec x = 1 / sin x

Let’s start the proof of the derivative of cot x

f(x) = cot x = cos(x)/sin(x)

u(x) = cos(x) and v(x)=sin(x)

u'(x) = -sin(x) and v'(x)=cos(x)

v²(x) = sin²(x)

f'(x) = {-sin(x).sin(x) - cos(x).cos(x)}/sin²(x)

f'(x) = -sin²(x)-cos²(x)/sin²(x)

f'(x) = -sin²(x)+cos²(x)/sin²(x)

By one of the trigonometric identities, cos²x + sin²x = 1.

f'(x) = - 1/ sin²(x)

d/dx cot(x) = -1 /sin²(x) = -cosec²(x)

Therefore, differentiation of cot x is -cosec²x.

Derivative of Cot x by Chain Rule

Assume y = cot x then we can write y = 1 / (tan x) = (tan x)^-1. Since we have power here, we can apply the power rule here. By power rule and chain rule,

y' = (-1) (tan x)^-2·d/dx (tan x)

The derivative of tan x is, d/dx (tan x) = sec²x

y= cot x

y' = -1/tan²x·(sec²x)

y' = - cot²x·sec²x

Now, cot x = (cos x)/(sin x) and sec x = 1/(cos x). So

y' = -(cos²x)/(sin²x) · (1/cos²x)

y' = -1/sin²x

Since, reciprocal of sin is cosec. i.e., 1/sin x = cosec x. So

y' = -cosec²x

Hence proved.

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Solved Examples on Derivative of Cot x

Example 1: Find the derivative of cot²x.

Solution:

Let f(x) = cot²x = (cot x)²

By using power rule and chain rule,

f'(x) = 2 cot x · d/dx(cot x)

We know that the derivative of cot x is -cosec²x. So

f'(x) = -2 cot x ·cosec²x

Example 2: Differentiate tan x with respect to cot x.

Solution:

Let v = tan x and u = cot x. Then dv/dx = sec²x and du/dx = -cosec²x.

We have to find dv/du. We can write this as

dv/du = (dv/dx) / (du/dx)

dv/du = (sec²x) / (-cosec²x)

dv/du = (1/cos²x) / (-1/sin²x)

dv/du = (-sin²x) / (cos²x)

dv/du = -tan²x

Example 3: Find the derivative of cot x · csc2x

Solution:

Let f(x) = cot x · cosec²x

By product rule,

f'(x) = cot x·d/dx (cosec²x) + cosec²x·d/dx(cot x)

f'(x) = cot x·(2 cosec x) d/dx (cosec x) + cosec²x (-cosec²x) (by chain rule)

f'(x) = 2 cosec x cot x (-cosec x cot x) - cosec⁴x

f'(x) = -2 cosec²x cot²x - cosec⁴x

Practice Questions on Derivative of Cot x

Various problems related to Derivative of Cot x are,

1. Find the derivative of 1/cot(x).

2. Calculate the derivative of cot(3x) + 2cot(x).

3. Determine the derivative of 1/cot(x)+1.

4. Determine the derivative of cot(x) - tan(x).

5. Determine the derivative of cot²(x).

Summary

To find the derivative of cot⁡(x), we use the quotient rule and trigonometric identities. The cotangent function cot⁡(x) is defined as ​. According to the quotient rule, the derivative of ​ is ​. Here, u=cos⁡(x) and v=sin⁡(x), with their derivatives being , respectively. Applying the quotient rule, we get , which simplifies to . Using the Pythagorean identity sin⁡²(x)+cos^⁡2(x)=1, the expression in the numerator simplifies to −1. Therefore, the derivative of cot(x) is ​, which is −csc⁡²(x).