121 LearnersLast updated on August 5th, 2025
Derivative of Cos 2x
We use the derivative of cos(2x), which is -2sin(2x), to understand how the cosine function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now discuss the derivative of cos(2x) in detail.
What is the Derivative of Cos 2x?
We now understand the derivative of cos 2x. It is commonly represented as d/dx (cos 2x) or (cos 2x)', and its value is -2sin(2x).
The function cos 2x has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below: Cosine Function: (cos(2x) is based on the cosine function).
Chain Rule: Rule for differentiating cos(2x) (since it involves differentiating a function within another function).
Sine Function: sin(x) is the derivative of cos(x).
Derivative of Cos 2x Formula
The derivative of cos 2x can be denoted as d/dx (cos 2x) or (cos 2x)'.
The formula we use to differentiate cos 2x is: d/dx (cos 2x) = -2sin(2x) (or) (cos 2x)' = -2sin(2x)
The formula applies to all x where the function is defined.
Proofs of the Derivative of Cos 2x
We can derive the derivative of cos 2x using proofs. To show this, we will use trigonometric identities along with the rules of differentiation.
There are several methods we use to prove this, such as: Using the Chain Rule Using the Product Rule We will now demonstrate that the differentiation of cos 2x results in -2sin(2x)
using the above-mentioned methods: Using Chain Rule To prove the differentiation of cos 2x using the chain rule, We use the formula: Cos 2x = cos(u) where u = 2x By chain rule: d/dx [cos(u)] = -sin(u) * du/dx
Let’s substitute u = 2x, d/dx (cos 2x) = -sin(2x) * (2) = -2sin(2x) Thus, the derivative of cos 2x is -2sin(2x).
Using Product Rule We will now prove the derivative of cos 2x using the product rule.
The step-by-step process is demonstrated below: Here, we use the formula, Cos 2x = cos(x + x) = cos(x)cos(x) - sin(x)sin(x)
Differentiating using the product rule: d/dx [cos(x)cos(x) - sin(x)sin(x)] = [cos(x)(-sin(x)) + cos(x)(-sin(x))] - [sin(x)cos(x) + sin(x)cos(x)] = -2sin(x)cos(x) - 2sin(x)cos(x) = -2sin(2x)
Thus, the derivative of cos 2x is -2sin(2x).
Higher-Order Derivatives of Cos 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 cos(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 cos(2x), we generally use fⁿ(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 is an integer multiple of π, the derivative is zero because sin(2x) is zero at those points. When x is 0, the derivative of cos 2x = -2sin(0), which is 0.
Common Mistakes and How to Avoid Them in Derivatives of Cos 2x
Students frequently make mistakes when differentiating cos 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 using the chain rule
Students may forget to apply the chain rule, which can lead to incomplete or incorrect results. They often skip steps and directly arrive at the result. 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 factor from the chain rule
They might not remember to multiply by the derivative of the inner function, which is 2 in the case of cos(2x). Keep in mind that you should consider the chain rule when differentiating composite functions. It will help you understand how each part of the function contributes to the derivative.
Mistake 3
Incorrect use of trigonometric identities
While differentiating functions such as cos(2x) using identities, students misapply the trigonometric identities.
For example: Incorrect differentiation: d/dx (cos(2x)) = -sin(2x). To avoid this mistake, write the identities correctly and apply them consistently.
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 placed before cos(2x).
For example, they incorrectly write d/dx (5 cos(2x)) = -sin(2x).
Students should check the constants in the terms and ensure they are multiplied properly.
For example, the correct equation is d/dx (5 cos(2x)) = -10sin(2x).
Mistake 5
Confusing cos with sin in the derivative
Students often confuse the derivative of the cosine function as being cosine itself.
For example: Incorrect: d/dx (cos(2x)) = cos(2x).
The derivative of cosine is negative sine. To fix this error, remember that the derivative of cos(x) is -sin(x) and apply this rule to cos(2x) as well.
Examples Using the Derivative of Cos 2x
Problem 1
Calculate the derivative of (cos 2x · sin(2x))
Here, we have f(x) = cos 2x · sin(2x). Using the product rule, f'(x) = u′v + uv′ In the given equation, u = cos 2x and v = sin(2x).
Let’s differentiate each term, u′ = d/dx (cos 2x) = -2sin(2x) v′ = d/dx (sin(2x)) = 2cos(2x)
Substituting into the given equation, f'(x) = (-2sin(2x))(sin(2x)) + (cos 2x)(2cos(2x))
Let’s simplify terms to get the final answer, f'(x) = -2sin²(2x) + 2cos²(2x)
Thus, the derivative of the specified function is -2sin²(2x) + 2cos²(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
A company manufactures gears, and the angle of rotation is represented by the function y = cos(2x) where y represents the position of a gear tooth at a distance x. If x = π/6 radians, measure the rate of change of the gear position.
We have y = cos(2x) (position of the gear)...(1) Now, we will differentiate the equation (1)
Take the derivative cos(2x): dy/dx = -2sin(2x) Given x = π/6 (substitute this into the derivative) dy/dx = -2sin(2(π/6)) dy/dx = -2sin(π/3) dy/dx = -2(√3/2) dy/dx = -√3
Hence, we get the rate of change of the gear position at x= π/6 as -√3.
Explanation
We find the rate of change of the gear position at x= π/6 as -√3, which means that at this point, the position of the gear is decreasing at a rate of √3 units per unit change in x.
Problem 3
Derive the second derivative of the function y = cos(2x).
The first step is to find the first derivative, dy/dx = -2sin(2x)...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [-2sin(2x)]
Here we use the chain rule, d²y/dx² = -2[cos(2x) * 2] d²y/dx² = -4cos(2x)
Therefore, the second derivative of the function y = cos(2x) is -4cos(2x).
Explanation
We use the step-by-step process, where we start with the first derivative. Using the chain rule, we differentiate -2sin(2x).
We then substitute the identity and simplify the terms to find the final answer.
Problem 4
Prove: d/dx (cos²(x)) = -2sin(x)cos(x).
Let’s start using the chain rule: Consider y = cos²(x) = [cos(x)]²
To differentiate, we use the chain rule: dy/dx = 2cos(x).d/dx [cos(x)]
Since the derivative of cos(x) is -sin(x), dy/dx = 2cos(x)(-sin(x)) dy/dx = -2sin(x)cos(x)
Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation.
Then, we replace cos(x) with its derivative. As a final step, we substitute y = cos²(x) to derive the equation.
Problem 5
Solve: d/dx (cos 2x/x)
To differentiate the function, we use the quotient rule: d/dx (cos 2x/x) = (d/dx (cos 2x)·x - cos 2x·d/dx(x))/x²
We will substitute d/dx (cos 2x) = -2sin(2x) and d/dx(x) = 1 = (-2sin(2x)·x - cos 2x·1)/x² = (-2x sin(2x) - cos 2x)/x²
Therefore, d/dx (cos 2x/x) = (-2x sin(2x) - cos 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 Cos 2x
1.Find the derivative of cos 2x.
2.Can we use the derivative of cos 2x in real life?
3.Is it possible to take the derivative of cos 2x at the point where x = π/2?
4.What rule is used to differentiate cos 2x/x?
5.Are the derivatives of cos 2x and cos⁻¹x the same?
6.Can we find the derivative of the cos 2x formula?
Important Glossaries for the Derivative of Cos 2x
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Cosine Function: A primary trigonometric function, written as cos(x).
- Sine Function: A trigonometric function that is the derivative of the cosine function, written as sin(x).
- Chain Rule: A rule for differentiating composite functions.
- Product Rule: A rule used to find the derivative of the product 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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