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111 LearnersLast updated on September 27, 2025
Derivative of sin²(2x)
We use the derivative of sin²(2x) to understand how this composite trigonometric function changes with respect to x. Derivatives help us in various real-life applications, including physics and engineering. We will investigate the derivative of sin²(2x) in detail.
What is the Derivative of sin²(2x)?
We now explore the derivative of sin²(2x). It is commonly represented as d/dx (sin²(2x)) or (sin²(2x))'. The derivative of sin²(2x) involves using the chain rule and the power rule. The function sin²(2x) is differentiable within its domain.
Key concepts include:
Sine Function: sin(2x) is the inner function.
Chain Rule: Used for differentiating composite functions like sin²(2x).
Power Rule: Used for the exponent part of the function.
Derivative of sin²(2x) Formula
The derivative of sin²(2x) can be denoted as d/dx (sin²(2x)). To find it, we apply the chain rule and the power rule: d/dx (sin²(2x)) = 2 * sin(2x) * d/dx(sin(2x))
Next, apply the chain rule on sin(2x): d/dx(sin(2x)) = 2 * cos(2x) Thus, the derivative is: d/dx (sin²(2x)) = 4 * sin(2x) * cos(2x)
Proofs of the Derivative of sin²(2x)
We can derive the derivative of sin²(2x) using proofs. This involves applying differentiation rules and trigonometric identities.
Here are a few methods:
- By Chain Rule
- By Trigonometric Identities
We will demonstrate that the differentiation of sin²(2x) results in 4 * sin(2x) * cos(2x) using these methods:
By Chain Rule
To prove using the chain rule, consider u = sin(2x), then y = u². Differentiate y = u² with respect to u: dy/du = 2u Differentiate u = sin(2x) with respect to x: du/dx = 2cos(2x) Using the chain rule: dy/dx = dy/du * du/dx = 2u * 2cos(2x) Substitute u = sin(2x): dy/dx = 2 * sin(2x) * 2cos(2x) = 4 * sin(2x) * cos(2x)
By Trigonometric Identities
We know sin²(2x) = (sin(2x))². Using the identity sin(2x) = 2sin(x)cos(x), we differentiate: d/dx (sin²(2x)) = d/dx (2sin(x)cos(x))² Using the product rule and chain rule, we find: d/dx (sin²(2x)) = 4sin(2x)cos(2x)
Higher-Order Derivatives of sin²(2x)
When a function is differentiated multiple times, the resulting derivatives are referred to as higher-order derivatives.
For sin²(2x), the first derivative is f′(x), indicating how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, denoted as f′′(x).
The third derivative, f′′′(x), follows from the second derivative, and this pattern continues. The nth derivative of sin²(2x) is generally denoted as fⁿ(x), representing the change in the rate of change.
Special Cases
When x is 0, the derivative of sin²(2x) = 4 * sin(0) * cos(0) = 0.
At points where sin(2x) = 0, such as x = π, 2π, ..., the derivative is also 0.
Common Mistakes and How to Avoid Them in Derivatives of sin²(2x)
Students frequently make mistakes when differentiating sin²(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 applying the chain rule correctly
Students may forget to apply the chain rule correctly when differentiating composite functions. This can lead to incorrect results.
Always identify the inner and outer functions and differentiate them separately before combining the results.
Mistake 2
Misapplying trigonometric identities
Students might not correctly apply trigonometric identities, leading to mistakes in differentiation.
It's essential to familiarize oneself with identities like sin(2x) = 2sin(x)cos(x) to simplify differentiation.
Mistake 3
Forgetting to multiply by the derivative of the inner function
When using the chain rule, students often forget to multiply by the derivative of the inner function. For example, they might write d/dx(sin²(2x)) = 2sin(2x) without multiplying by the derivative of sin(2x).
Always remember to include this important step.
Mistake 4
Ignoring special trigonometric points
Students might overlook special points where the function or its derivative has unique values, such as zeroes.
These points should be considered in analysis and problem-solving to avoid incorrect conclusions.
Mistake 5
Incorrect simplification of expressions
Simplifying expressions incorrectly can lead to errors.
Ensure every step in the simplification is accurate, especially when dealing with products or squares of trigonometric functions.
Examples Using the Derivative of sin²(2x)
Problem 1
Calculate the derivative of sin²(2x) · cos²(2x)
Here, we have f(x) = sin²(2x) · cos²(2x). Using the product rule, f'(x) = u′v + uv′ In the given equation, u = sin²(2x) and v = cos²(2x). Let’s differentiate each term, u′ = d/dx (sin²(2x)) = 4sin(2x)cos(2x) v′ = d/dx (cos²(2x)) = -4cos(2x)sin(2x) Substituting into the given equation, f'(x) = (4sin(2x)cos(2x)) · (cos²(2x)) + (sin²(2x)) · (-4cos(2x)sin(2x)) Simplifying terms, we get, f'(x) = 4sin(2x)cos³(2x) - 4sin³(2x)cos(2x) Thus, the derivative of the specified function is 4sin(2x)cos(2x)(cos²(2x) - sin²(2x)).
Explanation
We find the derivative of the given function by dividing it into two parts.
The first step is finding the derivative of each part and then combining them using the product rule to get the final result.
Problem 2
A pendulum swings with its angular position modeled by θ = sin²(2x), where x is time in seconds. Find the rate of change of the angle at x = π/6 seconds.
We have θ = sin²(2x) (angular position of the pendulum)...(1) Now, we will differentiate equation (1). Take the derivative of sin²(2x): dθ/dx = 4sin(2x)cos(2x) Given x = π/6, substitute this into the derivative: dθ/dx = 4sin(π/3)cos(π/3) Using values sin(π/3) = √3/2 and cos(π/3) = 1/2: dθ/dx = 4(√3/2)(1/2) = √3 Hence, the rate of change of the angle at x = π/6 seconds is √3.
Explanation
We find the rate of change of the pendulum’s angle at x = π/6 seconds by substituting the value into the derivative.
This gives the instantaneous rate of change of the angle with respect to time.
Problem 3
Derive the second derivative of the function y = sin²(2x).
The first step is to find the first derivative, dy/dx = 4sin(2x)cos(2x)...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [4sin(2x)cos(2x)] Using the product rule, d²y/dx² = 4[d/dx(sin(2x)cos(2x))] = 4[2cos²(2x) - 2sin²(2x)] = 8[cos²(2x) - sin²(2x)] Therefore, the second derivative of the function y = sin²(2x) is 8[cos²(2x) - sin²(2x)].
Explanation
We use the step-by-step process, starting with the first derivative.
Using the product rule, we differentiate sin(2x)cos(2x).
We simplify the terms to find the final answer for the second derivative.
Problem 4
Prove: d/dx (sin²(2x)) = 4sin(2x)cos(2x).
Let’s start using the chain rule: Consider y = sin²(2x). Let u = sin(2x), then y = u². To differentiate, use the chain rule: dy/dx = 2u * d/dx(u) Since d/dx(sin(2x)) = 2cos(2x), dy/dx = 2sin(2x) * 2cos(2x) = 4sin(2x)cos(2x) Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation.
We replace sin(2x) with u, differentiate, and substitute back to prove the result.
Problem 5
Solve: d/dx (sin²(2x)/ x)
To differentiate the function, we use the quotient rule: d/dx (sin²(2x)/ x) = (d/dx (sin²(2x)) * x - sin²(2x) * d/dx(x)) / x² Substitute d/dx(sin²(2x)) = 4sin(2x)cos(2x) and d/dx(x) = 1: = (4sin(2x)cos(2x) * x - sin²(2x)) / x² = (4xsin(2x)cos(2x) - sin²(2x)) / x² Therefore, d/dx (sin²(2x)/ x) = (4xsin(2x)cos(2x) - sin²(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 sin²(2x)
1.Find the derivative of sin²(2x).
2.Can we use the derivative of sin²(2x) in real life?
3.Is it possible to take the derivative of sin²(2x) at x = π?
4.What rule is used to differentiate sin²(2x)/x?
5.Are the derivatives of sin²(2x) and sin²(x) the same?
Important Glossaries for the Derivative of sin²(2x)
- Derivative: The derivative of a function indicates how the function changes with a small change in x.
- Chain Rule: A rule used to differentiate composite functions.
- Power Rule: A rule used to differentiate functions of the form u^n.
- Sine Function: A trigonometric function representing the y-coordinate of a point on the unit circle.
- Product Rule: A rule used to differentiate 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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