Introduction to the Derivative of csc2x
Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of csc 2x can be calculated by following the differentiation rules.
Or, we can directly find the derivative of csc (2x) by applying the first principle of differentiation. In this article, you will learn what the derivative of csc (2x) is and how to calculate the derivative of csc (2x) by using different approaches.
What is the derivative of csc(2x)?
The csc2x derivative with respect to x can be expressed as -2csc(2x)cot(2x), denoted as d/dx(csc(2x)). Understanding the relationship between the tangent and the cosecant functions is essential to computing this derivative. The tangent represents the slope of a line at a point of change in the function, and in a triangle, it is defined as the reciprocal of the sine function. Specifically, the cosecant of x can be written as 1/sin(x), providing a useful formula for calculating the derivative of csc2x.
Derivative of csc 2x formula
The formula of csc 2x differentiation is equal to the negative of the product of csc(2x) and cot(2x), which can be written as;
d/dx(csc(2x)) = -2csc(2x).cot(2x)
It represents the rate of change of csc 2x with respect to x. This formula is helpful in determining the slope of a line at a point of change in the csc(2x) function with respect to x.
How do you prove the derivative of csc2x?
There are different ways to derive derivatives csc (2x). Therefore, we can prove the derivative of csc (2x) by using;
- First Principle
- Chain Rule
- Quotient Rule
Each method provides a different way to compute the csc(2x) differentiation. By using these methods, we can mathematically prove the formula for finding csc2x derivatives.
Derivative of csc(2x) by first principle
According to the first principle of derivative calculator, the csc 2x derivative is equal to -csc(2x)cot(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 csc2x by first principle
To prove the derivative of csc (2x) by using the first principle, replace f(x) by csc (2x).
f'(x)=limh→0f(x+h)-f(x)/h
f'(x) = lim tan 2(x+h) - csc (2x)/h
Therefore,
f'(x) = lim [csc 2(x+h) - csc (2x)]/h
Now, by the trigonometric formula, csc x = 1/sin x. So,
f'(x) = lim [1/sin 2(x+h) - 1/sin (2x)]/h
f'(x) = lim [sin (2x) - sin 2(x+h)]/hsin 2(x+h) sin 2x
f'(x) = lim - [sin 2(x+h) - sin 2x]/hsin 2(x+h)sin 2x
Now,
f'(x) = - lim[sin 2(x+h) - sin 2x]/h. lim 1/[sin 2(x+h)sin 2x]
Since lim[sin 2(x+h) - sin 2x]/h = cos 2x
f'(x) = - cos (2x). lim 1/[sin 2(x+h)sin (2x)]
f'(x) = - cos (2x). 1/sin2 (2x)
Hence
f'(x) = - 2csc (2x).cot (2x)
The derivative of csc can also be verified by using first principle.
Derivative of csc(2x) by chain rule
To calculate the derivative of csc (2x), 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
Proof of derivative of csc 2x by chain rule
To prove the differentiation of csc 2x by using chain rule, we start by assuming that csc(2x) can be written as the combination of two functions. Using this let us find the csc2x derivative.
y = csc u where u = 2x
Using chain rule,
y = -csc u.cot u.du/dx
and
du/dx = 2
Now, using the value of u.
y = -2csc (2x).cot (2x)
Thus, we have derived the formula of the derivative of csc2x. The chain rule solver can be used to differentiate csc2x easily.
Derivative of cosecant 2x using quotient rule
Another method for finding the derivative of sec xtan 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 differentiation of csc 2x by quotient rule
To prove the derivative of csc (2x), we can start by writing it,
f(x) = csc (2x) = 1/sin (2x) =u/v
Supposing that u = 1 and v = sin (2x). Now by derivative quotient rule calculator,
f(x) = (vu - uv)/v2
f(x) = [sin (2x) d/dx(1) - d/dx(sin (2x))] / (cos 3x)2
= [0 - (2cos (2x))] / sin2 2x
= [-2cos (2x)]/ sin2 2x
= -2csc (2x).cot (2x)
Hence, we have derived the derivative of csc (2x) using the quotient rule of differentiation.
How to find the derivative of csc 2x with a calculator?
The easiest way to calculate the derivative of csc (2x) is by using an online tool. You can use our differentiate calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.
- Write the function as csc(2x)in the enter function box. In this step, you need to provide input value as a function as you have to differentiate cosec x.
- Now, select the variable by which you want to differentiate csc(2x). Here you have to choose x.
- Select how many times you want to differentiate csc(2x). In this step, you can choose 2 for second derivative test, 3 for third derivative and so on.
- Click on the calculate button.
After completing these steps, you will receive the csc2x derivative within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.
Frequently asked questions
What is the derivative of csc u?
The derivative of cosec u is negative of the product of trigonometric functions cosec u and cot u, that is, -cosec u cot u. The differentiation of csc u is the process of evaluating the derivative of cosec x with respect to angle u.
What is csc equal to?
The cosecant is a trigonometric function which is represented as csc. It is the reciprocal of sine function and written as csc x = 1/sin x.
What is the derivative of sec?
The derivative of secx formula with respect to x is written as d/dx(sec x) and it is equal to sec x tan x. i.e., the differentiation of sec x is the product of sec x and tan x.