109 LearnersLast updated on August 5th, 2025
Derivative of csc(2x)
We use the derivative of csc(2x), which is -2csc(2x)cot(2x), to understand how the cosecant function changes in response to a slight change in x. Derivatives help us calculate rates of change and other practical applications. We will now discuss the derivative of csc(2x) in detail.
What is the Derivative of csc(2x)?
We now understand the derivative of csc(2x). It is commonly represented as d/dx (csc(2x)) or (csc(2x))', and its value is -2csc(2x)cot(2x). The function csc(2x) has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Cosecant Function: (csc(x) = 1/sin(x)).
Chain Rule: Rule for differentiating csc(2x) (since it involves a composite function).
Cotangent Function: cot(x) = 1/tan(x).
Derivative of csc(2x) Formula
The derivative of csc(2x) can be denoted as d/dx (csc(2x)) or (csc(2x))'. The formula we use to differentiate csc(2x) is: d/dx (csc(2x)) = -2csc(2x)cot(2x) The formula applies to all x where sin(2x) ≠ 0
Proofs of the Derivative of csc(2x)
We can derive the derivative of csc(2x) using proofs. To show this, we will use trigonometric identities along with differentiation rules. There are several methods we use to prove this, such as:
- By First Principle
- Using Chain Rule
- Using Product Rule
We will now demonstrate that the differentiation of csc(2x) results in -2csc(2x)cot(2x) using the above-mentioned methods: Using Chain Rule
To prove the differentiation of csc(2x) using the chain rule, we use the formula: Csc(2x) = 1/sin(2x)
Let u = 2x, then csc(u) = 1/sin(u)
By the chain rule: d/dx [csc(u)] = -csc(u)cot(u) · du/dx Let’s substitute u = 2x, so du/dx = 2 d/dx (csc(2x)) = -csc(2x)cot(2x) · 2 = -2csc(2x)cot(2x)
Thus, the derivative of csc(2x) is -2csc(2x)cot(2x).
Higher-Order Derivatives of csc(2x)
When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.
To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like csc(2x).
For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(x). Similarly, the third derivative, f′′′(x), is the result of the second derivative and this pattern continues.
For the nth Derivative of csc(2x), we generally use f⁽ⁿ⁾(x) for the nth derivative of a function f(x), which tells us the change in the rate of change (continuing for higher-order derivatives).
Special Cases:
When x equals any integer multiple of π/2, the derivative is undefined because csc(2x) has vertical asymptotes there. When x equals 0, the derivative of csc(2x) = -2csc(0)cot(0), which does not exist since csc(0) is undefined.
Common Mistakes and How to Avoid Them in Derivatives of csc(2x)
Students frequently make mistakes when differentiating csc(2x). These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:
Mistake 1
Not simplifying the equation
Students may forget to simplify the equation, which can lead to incomplete or incorrect results. They often skip steps and directly arrive at the result, especially when solving using the chain rule. Ensure that each step is written in order. Students might think it is awkward, but it is important to avoid errors in the process.
Mistake 2
Forgetting the Undefined Points of csc(2x)
They might not remember that csc(2x) is undefined at points where sin(2x) = 0. Keep in mind that you should consider the domain of the function that you differentiate. It will help you understand that the function is not continuous at such certain points.
Mistake 3
Incorrect use of Chain Rule
While differentiating functions such as csc(2x), students might misapply the chain rule. For example: Incorrect differentiation: d/dx (csc(2x)) = -csc(2x)cot(2x) without considering the derivative of the inner function. Correct application: d/dx (csc(2x)) = -2csc(2x)cot(2x). Always check for errors in the calculation and ensure it is properly simplified.
Mistake 4
Not writing Constants and Coefficients
There is a common mistake that students at times forget to multiply the constants associated with the argument of the function. For example, they incorrectly write d/dx (csc(2x)) = -csc(2x)cot(2x) instead of -2csc(2x)cot(2x). Students should check the constants in the terms and ensure they are multiplied properly.
Mistake 5
Not Applying the Chain Rule
Students often forget to use the chain rule. This happens when the derivative of the inner function is not considered. For example: Incorrect: d/dx (csc(2x)) = -csc(2x)cot(2x). To fix this error, students should divide the functions into inner and outer parts. Then, make sure that each function is differentiated. For example, d/dx (csc(2x)) = -2csc(2x)cot(2x).
Examples Using the Derivative of csc(2x)
Problem 1
Calculate the derivative of (csc(2x)·cot(2x))
Here, we have f(x) = csc(2x)·cot(2x).
Using the product rule, f'(x) = u′v + uv′ In the given equation, u = csc(2x) and v = cot(2x).
Let’s differentiate each term, u′= d/dx (csc(2x)) = -2csc(2x)cot(2x) v′= d/dx (cot(2x)) = -2csc²(2x)
Substituting into the given equation, f'(x) = (-2csc(2x)cot(2x))·cot(2x) + csc(2x)·(-2csc²(2x))
Let’s simplify terms to get the final answer, f'(x) = -2csc(2x)cot²(2x) - 2csc³(2x)
Thus, the derivative of the specified function is -2csc(2x)cot²(2x) - 2csc³(2x).
Explanation
We find the derivative of the given function by dividing the function into two parts. The first step is finding its derivative and then combining them using the product rule to get the final result.
Problem 2
The elevation of a hill is represented by the function y = csc(2x) where y represents the height at a distance x from the base. If x = π/6 meters, measure the rate of change of elevation of the hill.
We have y = csc(2x) (elevation of the hill)...(1)
Now, we will differentiate the equation (1) Take the derivative csc(2x): dy/dx = -2csc(2x)cot(2x)
Given x = π/6 (substitute this into the derivative)
dy/dx = -2csc(π/3)cot(π/3) csc(π/3) = 2/√3 and cot(π/3) = 1/√3 dy/dx = -2(2/√3)(1/√3) = -4/3
Hence, the rate of change of elevation of the hill at a distance x = π/6 is -4/3.
Explanation
We find the rate of change of elevation at x = π/6 as -4/3, indicating the elevation decreases at this rate with respect to the horizontal distance.
Problem 3
Derive the second derivative of the function y = csc(2x).
The first step is to find the first derivative, dy/dx = -2csc(2x)cot(2x)...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [-2csc(2x)cot(2x)]
We use the product rule, d²y/dx² = -2[d/dx (csc(2x))cot(2x) + csc(2x)d/dx (cot(2x))] = -2[-2csc(2x)cot²(2x) - 2csc³(2x)]
Therefore, the second derivative of the function y = csc(2x) is 4csc(2x)cot²(2x) + 4csc³(2x).
Explanation
We use the step-by-step process, where we start with the first derivative. Using the product rule, we differentiate -2csc(2x)cot(2x). We then simplify the terms to find the final answer.
Problem 4
Prove: d/dx (csc²(2x)) = -4csc²(2x)cot(2x).
Let’s start using the chain rule: Consider y = csc²(2x) [csc(2x)]²
To differentiate, we use the chain rule: dy/dx = 2csc(2x)·d/dx [csc(2x)]
Since the derivative of csc(2x) is -2csc(2x)cot(2x), dy/dx = 2csc(2x)·(-2csc(2x)cot(2x)) = -4csc²(2x)cot(2x)
Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace csc(2x) with its derivative. As a final step, we substitute y = csc²(2x) to derive the equation.
Problem 5
Solve: d/dx (csc(2x)/x)
To differentiate the function, we use the quotient rule: d/dx (csc(2x)/x) = (d/dx (csc(2x))·x - csc(2x)·d/dx(x))/x²
We will substitute d/dx (csc(2x)) = -2csc(2x)cot(2x) and d/dx (x) = 1 = (-2csc(2x)cot(2x)·x - csc(2x)·1) / x² = (-2x csc(2x)cot(2x) - csc(2x)) / x²
Therefore, d/dx (csc(2x)/x) = (-2x csc(2x)cot(2x) - csc(2x)) / x²
Explanation
In this process, we differentiate the given function using the quotient rule. As a final step, we simplify the equation to obtain the final result.
FAQs on the Derivative of csc(2x)
1.Find the derivative of csc(2x).
2.Can we use the derivative of csc(2x) in real life?
3.Is it possible to take the derivative of csc(2x) at the point where x = π/2?
4.What rule is used to differentiate csc(2x)/x?
5.Are the derivatives of csc(2x) and csc(x) the same?
Important Glossaries for the Derivative of csc(2x)
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Cosecant Function: A trigonometric function that is the reciprocal of the sine function, represented as csc(x).
- Cotangent Function: A trigonometric function that is the reciprocal of the tangent function, represented as cot(x).
- Chain Rule: A fundamental rule in calculus used to differentiate composite functions.
- Quotient Rule: A rule used to find the derivative of the division of two functions.
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Jaskaran Singh Saluja
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Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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