Introduction to the derivative of cot2x
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 cot 2x derivative by applying the first principle of differentiation. In this article, you will learn what the cot2x formula of derivative is and how to calculate the differentiation of cot 2x by using different approaches.
What is the cot2x differentiation?
The derivative of cot(2x) with respect to the variable x is given by the square of the negative cosecant of x, represented as d/dx(cot(2x)). This expression represents the rate of change of the trigonometric function cot(2x). In a triangle, cot(2x) is defined as the ratio of the cosine and sine functions and can be written as;
cot 2x=cos 2x/sin 2x
By differentiating cot(2x) using the appropriate formula, you can find its instantaneous rate of change at any given value of x.
Derivative of cot 2x formula
The derivative of cot2x formula is equal to the negative of the cosecant squared function, represented as;
d/dx(cot(2x)) = -cosec(2x)^2
This expression denotes the instantaneous rate of change of the trigonometric function cot(2x) with respect to the variable x. It is essential in calculus and trigonometry for determining slopes, rates of change, and other related quantities.
How do you prove the cot 2x formula for derivatives?
There are several methods to derive the differentiation of cot2x. Three commonly used methods are;
- First Principle
- Chain Rule
- Quotient Rule
Each method provides a different way to compute the csc 2x derivative. By using these methods, we can mathematically prove the formula for finding derivatives of csc2(x).
Derivative of cot2x by first principle
According to the first principle of derivative, the cot 2x derivative is equal to -2csc^2(2x). The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,
f(x)=lim f(x+h)-f(x) / h
This formula allows us to determine the rate of change of a function at a specific point by using limit definition of derivative.
Proof of derivative of cot 2x by first principle
To prove the cot2x differentiation by using first principle, replace f(x) by cot(2x).
f′(x)=limh→0f(x+h)-f(x)/h
f'(x) = lim cot 2(x+h) - cot(2x)/h
Since cot(2x) = cos 2x/sin 2x, therefore,
f'(x) = lim cos 2(x+h) /sin 2(x+h) - cos 2x/sin 2x/h
More simplification,
f'(x) = lim sin 2x cos 2(x+h)- sin 2(x+h) cos 2x/hsin 2x.sin 2(x+h)
Now, by the trigonometric formula, sin A cos B - cos A sin B = sin (A - B)
f'(x) = lim sin 2(x-(x+h))/hsin 2x.sin 2(x+h)
f'(x) = lim sin 2(-h)/hsin 2x.sin 2(x+h)
We have sin(-h) = -sin h, therefore,
f'(x) = -lim sin 2h/ 2h * lim 2/[sin 2x. sin 2(x+h)]
When h approaches to zero,
f'(x) = - 1 * 1/sin 2x. sin 2x
Also the slope of a curve calculator helps you to find the slope of a curve. Now, we can write the differentiation of cot2x as;
f'(x) = -2cosec2 2x
Derivative of cot2x by chain rule
To calculate the derivative of cot 2x formula, 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 derivative is defined as;
dy / dx = dy / du x du / dx
Proof of cot2x differentiation by chain rule
To prove the cot 2x differentiation by using chain rule, we can start by assuming that,
f(x) = cot(2x) = 1/tan 2x = (tan 2x)-1
By using chain rule of differentiation,
f'(x) = (-2) (tan 2x)-2 d/dx (sec2 2x)
Simplifying,
f'(x) = (-2/tan2 2x) . (sec2 2x)
Again,
f'(x) = -2sec2 2x/ tan2 2x
Since sin x / cos x =tan x and 1/cos x = sec x, therefore we have
f'(x) = -2cosec2 2x
You can also use derivative chain rule calculator to differentiate cot2x with some clicks.
Differentiation of cot2x using quotient rule
Another method for finding the derivative of sec x tan x is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. 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 derivative of cot2x by quotient rule
To prove the cot 2x derivative, we can write it,
f(x) = cot(2x)=1/ tan x =u/v
Supposing that u = 1 and v = tan x. Now by using derivative quotient rule calculator,
f x) = (vu - uv /v2
f(x) = [tan x d/dx(1) - 1. d/dx(tan x)] / (tan x)2
= [tan x (0) - 1 (sec2 x)] / tan2x
= (-sec2 x ) / tan2x
= -cosec2 x
Hence, we have derived the cot2x differentiation using the quotient rule of differentiation. The cot2x derivative can also be calculated by using product rule differentiation.
How to find the differentiation of cot2x with a calculator?
The easiest way to calculate the differentiation 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 dy/dx calculator.
- 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 cot 2x formula of derivative.
- Now, select the variable by which you want to differentiate cot2x. Here you have to choose x
- Select how many times you want to differentiate cotangent x. In this step, you can choose 2 for second, 3 for triple derivative and so on.
- Click on the calculate button. After this step, you will get the derivative of cot 2x within a few seconds.
After completing these steps, you will receive the cot2x differentiation within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.
Frequently asked questions
What is cot 2x?
Cot2x is an important formula in trigonometry which is obtained by multiplying the angle by 2. It is used to calculate cotangent function. It can be also expressed by using sine and cosine functions.
What is cot(2x) equal to?
The cot(2x) 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(2x) = cos x/ sin x
What is the second derivative of cot2x?
The first cot2x derivative is equal to the negative of the square of cosec x. And the second derivative of cot can be calculated by differentiating the first derivative. Therefore, the second derivative of cot 2x is;
d2/dx2 (cot(2x)) = 2cot(2x) cosec2 2x;