Last updated on July 24th, 2025
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.
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.
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.
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 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.
When x = π/6, the derivative is -3sin(π/2) = -3. When x = 0, the derivative of cos(3x) = -3sin(0) = 0.
Mistakes in differentiating cos(3x) often arise from misapplying rules. Here are common errors and solutions:
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).
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.
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).
The rate of temperature change is found by substituting x = 2 into the derivative, which gives the rate of change at that specific time.
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).
We start with the first derivative and then differentiate it to obtain the second derivative using the product rule and trigonometric identities.
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.
Using the chain rule, we differentiate the function by identifying the inner and outer functions, then multiply by the derivative of the inner function.
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.
We apply the quotient rule to differentiate the given function, simplifying the equation to reach the final result.
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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