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.
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.
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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
Some examples related to Derivative of Cot x are,
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,
Q1. Find the derivative of 1/cot(x).
Q2. Calculate the derivative of cot(3x) + 2cot(x).
Q3. Determine the derivative of 1/cot(x)+1.
Q4. Determine the derivative of cot(x) – tan(x).
Q5. Determine the derivative of cot2(x).
Derivative of Cot x – FAQs
What is Derivative?
The derivative of the function is defined as the rate of change of the function with respect to a independent variable.
What is Formula for Derivative of Cot x?
The formula for derivative of cot x is: (d/dx) cot x = -cosec2x
What is Derivative of Cot (-x)?
Derivative of cot (-x) is cosec2(-x).
What are Different Methods to Prove Derivative of Cot x?
The different methods to prove derivative of cot x are:
- By using First Principle of Derivative
- By Quotient Rule
- By Chain Rule
What is Derivative of cot t?
The derivative of cot t is (-cosec2t)