116 LearnersLast updated on August 5th, 2025
Derivative of cos3x
We use the derivative of cos(3x) to understand how the cosine function changes with respect to slight changes in x. Derivatives play a crucial role in various real-life applications, such as calculating rates of change. We will now explore the derivative of cos(3x) in detail.
What is the Derivative of cos3x?
The derivative of cos(3x) is commonly represented as d/dx (cos(3x)) or (cos(3x))'. Using the chain rule, its derivative is -3sin(3x). This reflects how the function changes as x varies within its domain. Here are key concepts to understand: Cosine Function: cos(x) represents the cosine function. Chain Rule: Used for differentiating composite functions like cos(3x). Sine Function: sin(x), which is used in the derivative of cosine.
Derivative of cos3x Formula
The derivative of cos(3x) can be expressed as d/dx (cos(3x)) or (cos(3x))'. The formula is: d/dx (cos(3x)) = -3sin(3x) This formula applies to all x where the function is defined.
Proofs of the Derivative of cos3x
We can derive the derivative of cos(3x) using various methods. Let's explore these methods: By First Principle Using Chain Rule By First Principle The derivative of cos(3x) using the First Principle involves expressing the derivative as a limit. Consider f(x) = cos(3x). The derivative is expressed as: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Substituting f(x) = cos(3x), we have: f'(x) = limₕ→₀ [cos(3(x + h)) - cos(3x)] / h Using the identity cos A - cos B = -2sin((A + B)/2)sin((A - B)/2), we simplify: f'(x) = limₕ→₀ [-2sin((3x + 3h + 3x)/2)sin((3h)/2)] / h = limₕ→₀ [-2sin(3x + 3h/2)sin(3h/2) / h] Using limit properties, f'(x) = -3sin(3x). Using Chain Rule For the chain rule proof, consider y = cos(3x). Let u = 3x, then y = cos(u). The derivative dy/du = -sin(u), and du/dx = 3. Applying the chain rule, dy/dx = dy/du * du/dx = -sin(3x) * 3 = -3sin(3x).
Higher-Order Derivatives of cos3x
Higher-order derivatives are obtained by differentiating a function multiple times. For example, the second derivative provides insight into the curvature of the function. For cos(3x), the first derivative is -3sin(3x). The second derivative, derived from the first derivative, is: d²y/dx² = d/dx(-3sin(3x)) = -3 * 3cos(3x) = -9cos(3x). Higher-order derivatives continue this pattern.
Special Cases:
When x = π/6, the derivative is -3sin(π/2) = -3. When x = 0, the derivative of cos(3x) = -3sin(0) = 0.
Common Mistakes and How to Avoid Them in Derivatives of cos3x
Mistakes in differentiating cos(3x) often arise from misapplying rules. Here are common errors and solutions:
Mistake 1
Ignoring the Chain Rule
Students may forget to apply the chain rule when differentiating a composite function like cos(3x). Always identify the inner function and apply the chain rule to avoid errors.
Mistake 2
Incorrect Trigonometric Identities
Errors can occur when using incorrect trigonometric identities. For example, using the wrong identity for sin or cos can lead to incorrect derivatives. Verify identities before use.
Mistake 3
Forgetting the Multiplier
When applying the chain rule, forgetting to multiply by the derivative of the inner function (3 in cos(3x)) is a common mistake. Always include this multiplier in your calculations.
Mistake 4
Misapplying the First Principle
Students may incorrectly apply the limit process in the First Principle. Follow each step carefully and ensure correct use of trigonometric identities to avoid errors.
Mistake 5
Not Recognizing Special Angles
When x is a special angle, such as π/6 or 0, ensure you substitute correctly to find the derivative value. Remember the trigonometric values at these points for accurate results.
Examples Using the Derivative of cos3x
Problem 1
Calculate the derivative of (cos3x * sinx)
Let's use the product rule for differentiation. f(x) = cos(3x) * sin(x) Using the product rule, f'(x) = u′v + uv′ where u = cos(3x) and v = sin(x). Differentiate each term: u′ = d/dx(cos(3x)) = -3sin(3x) v′ = d/dx(sin(x)) = cos(x) Substitute into the product rule: f'(x) = (-3sin(3x))sin(x) + (cos(3x))(cos(x)) f'(x) = -3sin(3x)sin(x) + cos(3x)cos(x) Thus, the derivative of the specified function is -3sin(3x)sin(x) + cos(3x)cos(x).
Explanation
We apply the product rule by dividing the function into two parts, finding the derivative of each, and then using the product rule to obtain the final result.
Problem 2
A company monitors temperature changes using the function T = cos(3x), where T is the temperature in degrees and x is the time in hours. Calculate the rate of temperature change at x = 2 hours.
Given T = cos(3x), the rate of change of temperature is the derivative: dT/dx = -3sin(3x). For x = 2, dT/dx = -3sin(3*2) = -3sin(6). Hence, the rate of temperature change at x = 2 hours is -3sin(6).
Explanation
The rate of temperature change is found by substituting x = 2 into the derivative, which gives the rate of change at that specific time.
Problem 3
Derive the second derivative of the function y = cos(3x).
First, find the first derivative: dy/dx = -3sin(3x). Now differentiate again to get the second derivative: d²y/dx² = d/dx(-3sin(3x)) = -3 * 3cos(3x) = -9cos(3x). Therefore, the second derivative is -9cos(3x).
Explanation
We start with the first derivative and then differentiate it to obtain the second derivative using the product rule and trigonometric identities.
Problem 4
Prove: d/dx (cos²(3x)) = -6cos(3x)sin(3x).
Use the chain rule: Let y = cos²(3x) = [cos(3x)]². Differentiate using the chain rule: dy/dx = 2cos(3x) * d/dx(cos(3x)) = 2cos(3x) * (-3sin(3x)) = -6cos(3x)sin(3x). Hence proved.
Explanation
Using the chain rule, we differentiate the function by identifying the inner and outer functions, then multiply by the derivative of the inner function.
Problem 5
Solve: d/dx (cos(3x)/x)
Apply the quotient rule: d/dx (cos(3x)/x) = (d/dx (cos(3x)) * x - cos(3x) * d/dx(x)) / x² = [-3sin(3x) * x - cos(3x) * 1] / x² = [-3xsin(3x) - cos(3x)] / x². Therefore, the derivative is [-3xsin(3x) - cos(3x)] / x².
Explanation
We apply the quotient rule to differentiate the given function, simplifying the equation to reach the final result.
FAQs on the Derivative of cos3x
1.Find the derivative of cos(3x).
2.Can we use the derivative of cos(3x) in real life?
3.Is it possible to take the derivative of cos(3x) at the point where x = π/2?
4.What rule is used to differentiate cos(3x)/x?
5.Are the derivatives of cos(3x) and cos⁻¹x the same?
Important Glossaries for the Derivative of cos3x
Derivative: Indicates how a function changes as its input changes. Cosine Function: A trigonometric function represented as cos(x). Chain Rule: A rule for differentiating composite functions. Sine Function: A trigonometric function, sin(x), used in derivative calculations. Quotient Rule: A rule for differentiating the quotient 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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