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Last updated on August 5th, 2025

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Derivative of Sin(6x)

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We use the derivative of sin(6x), which is 6cos(6x), as a tool for understanding how the sine function changes in response to a slight change in x. Derivatives help us calculate various rates of change in real-life situations. We will now discuss the derivative of sin(6x) in detail.

Derivative of Sin(6x) for US Students
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What is the Derivative of Sin(6x)?

We now understand the derivative of sin(6x). It is commonly represented as d/dx (sin(6x)) or (sin(6x))', and its value is 6cos(6x). The function sin(6x) has a clearly defined derivative, indicative of its differentiability within its domain. The key concepts are mentioned below: Sine Function: (sin(6x) is a transformation of the basic sine function). Chain Rule: Rule for differentiating composite functions like sin(6x). Cosine Function: cos(x), related as the derivative of the sine function.

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Derivative of Sin(6x) Formula

The derivative of sin(6x) can be denoted as d/dx (sin(6x)) or (sin(6x))'. The formula we use to differentiate sin(6x) is: d/dx (sin(6x)) = 6cos(6x) The formula applies to all x.

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Proofs of the Derivative of Sin(6x)

We can derive the derivative of sin(6x) 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: By First Principle Using Chain Rule By First Principle The derivative of sin(6x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of sin(6x) using the first principle, we will consider f(x) = sin(6x). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = sin(6x), we write f(x + h) = sin(6(x + h)). Substituting these into equation (1), f'(x) = limₕ→₀ [sin(6(x + h)) - sin(6x)] / h = limₕ→₀ [2cos(3(2x + h))sin(3h)] / h Using the limit formula, limₕ→₀ (sin h)/ h = 1, f'(x) = 6cos(6x) Hence, proved. Using Chain Rule To prove the differentiation of sin(6x) using the chain rule, Consider f(x) = sin(u) where u = 6x. By chain rule: d/dx [sin(u)] = cos(u) * du/dx. Here, du/dx = 6. Thus, d/dx [sin(6x)] = cos(6x) * 6 = 6cos(6x).

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Higher-Order Derivatives of Sin(6x)

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 sin(6x). 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 sin(6x), 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).

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Special Cases:

When x is a multiple of π, the derivative is 0 because cos(6x) is 0 at those points. When x is 0, the derivative of sin(6x) = 6cos(0), which is 6.

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Common Mistakes and How to Avoid Them in Derivatives of Sin(6x)

Students frequently make mistakes when differentiating sin(6x). These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:

Mistake 1

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Not applying the Chain Rule

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Students often forget to use the chain rule. This happens when they do not consider the derivative of the inner function (6x). For example: Incorrect: d/dx (sin(6x)) = cos(6x). To fix this error, students should recognize the function as composite, with an inner function and apply the chain rule: d/dx (sin(6x)) = 6cos(6x).

Mistake 2

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Confusing Sine and Cosine Derivatives

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Students may confuse the derivatives of sine and cosine. Remember: the derivative of sin(u) is cos(u), while the derivative of cos(u) is -sin(u). In the context of sin(6x), d/dx (sin(6x)) = 6cos(6x).

Mistake 3

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Neglecting Constants in the Derivative

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A common error is neglecting to multiply by the constant from the inner function during differentiation. For example, students might incorrectly write d/dx (sin(6x)) = cos(6x) instead of 6cos(6x). Always remember to multiply by the derivative of the inner function.

Mistake 4

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Incorrectly Assuming Derivative is the Same as Original Function

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Sometimes, students assume the derivative of sin(6x) is another sine function due to oversight. It is essential to apply differentiation rules correctly and not assume the function remains the same.

Mistake 5

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Forgetting Special Values of Trigonometric Functions

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Students may forget specific values of trigonometric functions at key points, such as cos(0) = 1. This can lead to incorrect calculations of the derivative at specific points. Remember to use known values of trigonometric functions to simplify your results.

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Examples Using the Derivative of Sin(6x)

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Problem 1

Calculate the derivative of (sin(6x)·cos(6x))

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Here, we have f(x) = sin(6x)·cos(6x). Using the product rule, f'(x) = u′v + uv′ In the given equation, u = sin(6x) and v = cos(6x). Let’s differentiate each term, u′ = d/dx (sin(6x)) = 6cos(6x) v′ = d/dx (cos(6x)) = -6sin(6x) Substituting into the given equation, f'(x) = (6cos(6x))(cos(6x)) + (sin(6x))(-6sin(6x)) = 6cos²(6x) - 6sin²(6x) Thus, the derivative of the specified function is 6cos²(6x) - 6sin²(6x).

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.

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Problem 2

A pendulum swings such that its angle θ with the vertical is given by θ = sin(6t). At t = π/6 seconds, find the rate of change of the angle.

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We have θ = sin(6t) (angle of the pendulum)...(1) Now, we will differentiate the equation (1) with respect to time t. dθ/dt = 6cos(6t) Given t = π/6, substitute this into the derivative: dθ/dt = 6cos(6π/6) dθ/dt = 6cos(π) = 6(-1) = -6 Hence, the rate of change of the angle at t = π/6 seconds is -6.

Explanation

We find the rate of change of the angle at t = π/6 seconds, which means the angle is decreasing at a rate of 6 units per second at this point.

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Problem 3

Derive the second derivative of the function y = sin(6x).

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The first step is to find the first derivative, dy/dx = 6cos(6x)...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [6cos(6x)] Using the chain rule, d²y/dx² = 6(-6sin(6x)) = -36sin(6x) Therefore, the second derivative of the function y = sin(6x) is -36sin(6x).

Explanation

We use the step-by-step process, where we start with the first derivative. Using the chain rule, we differentiate 6cos(6x). We simplify the terms to find the final answer.

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Problem 4

Prove: d/dx (sin²(6x)) = 12sin(6x)cos(6x).

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Let’s start using the chain rule: Consider y = sin²(6x) = [sin(6x)]² To differentiate, we use the chain rule: dy/dx = 2sin(6x)·d/dx [sin(6x)] Since the derivative of sin(6x) is 6cos(6x), dy/dx = 2sin(6x)·6cos(6x) = 12sin(6x)cos(6x) Hence proved.

Explanation

In this step-by-step process, we used the chain rule to differentiate the equation. As a final step, we substitute the derivative of sin(6x) to derive the equation.

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Problem 5

Solve: d/dx (sin(6x)/x)

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To differentiate the function, we use the quotient rule: d/dx (sin(6x)/x) = (d/dx (sin(6x))·x - sin(6x)·d/dx(x))/x² We will substitute d/dx (sin(6x)) = 6cos(6x) and d/dx (x) = 1 = (6cos(6x)·x - sin(6x)·1) / x² = (6xcos(6x) - sin(6x)) / x² Therefore, d/dx (sin(6x)/x) = (6xcos(6x) - sin(6x)) / 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.

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FAQs on the Derivative of Sin(6x)

1.Find the derivative of sin(6x).

Using the chain rule on sin(6x) gives 6cos(6x).

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2.Can we use the derivative of sin(6x) in real life?

Yes, we can use the derivative of sin(6x) in real life to analyze oscillations, waves, and other phenomena in fields like physics and engineering.

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3.Is it possible to take the derivative of sin(6x) at any point?

Yes, sin(6x) is differentiable at any point since it is a continuous trigonometric function.

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4.What rule is used to differentiate sin(6x)/x?

We use the quotient rule to differentiate sin(6x)/x, d/dx (sin(6x)/x) = (6xcos(6x) - sin(6x)) / x².

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5.Are the derivatives of sin(6x) and sin⁻¹(6x) the same?

No, they are different. The derivative of sin(6x) is 6cos(6x), while the derivative of sin⁻¹(6x) is 1/√(1-(6x)²) * 6.

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Important Glossaries for the Derivative of Sin(6x)

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Sine Function: A trigonometric function that describes a smooth periodic oscillation, which is often written as sin(x). Cosine Function: A trigonometric function that is the derivative of the sine function, represented as cos(x). Chain Rule: A rule for differentiating composite functions like sin(6x). Product Rule: A rule used to find the derivative of the product of two functions.

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Jaskaran Singh Saluja

About the Author

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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Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

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