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109 LearnersLast updated on 10 September 2025
Derivative of sin(4x)
We use the derivative of sin(4x), which is 4cos(4x), as a measuring tool for how the sine function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now talk about the derivative of sin(4x) in detail.
What is the Derivative of sin(4x)?
We now understand the derivative of sin(4x). It is commonly represented as d/dx (sin(4x)) or (sin(4x))', and its value is 4cos(4x). The function sin(4x) has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Sine Function: (sin(x) is a periodic function).
Chain Rule: Rule for differentiating composite functions like sin(4x).
Cosine Function: cos(x) is the derivative of sin(x).
Derivative of sin(4x) Formula
The derivative of sin(4x) can be denoted as d/dx (sin(4x)) or (sin(4x))'.
The formula we use to differentiate sin(4x) is: d/dx (sin(4x)) = 4cos(4x)
The formula applies to all x.
Proofs of the Derivative of sin(4x)
We can derive the derivative of sin(4x) using proofs. To show this, we will use the 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
We will now demonstrate that the differentiation of sin(4x) results in 4cos(4x) using the above-mentioned methods:
By First Principle
The derivative of sin(4x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.
To find the derivative of sin(4x) using the first principle, we will consider f(x) = sin(4x). 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(4x), we write f(x + h) = sin(4(x + h)).
Substituting these into equation (1), f'(x) = limₕ→₀ [sin(4(x + h)) - sin(4x)] / h = limₕ→₀ [ 2cos(4x + 2h)sin(2h) ] / h Using limit formulas, limₕ→₀ (sin h)/ h = 1. f'(x) = 4cos(4x) Hence, proved.
Using Chain Rule
To prove the differentiation of sin(4x) using the chain rule, We use the formula: Let u = 4x Then, d/dx (u) = 4 So, d/dx (sin(u)) = cos(u) · d/dx (u)
Substituting back, d/dx (sin(4x)) = cos(4x) · 4 = 4cos(4x)
Higher-Order Derivatives of sin(4x)
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(4x).
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(4x), 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/4, the derivative is defined and will be a specific value based on the cosine function. When x = 0, the derivative of sin(4x) = 4cos(0), which is 4.
Common Mistakes and How to Avoid Them in Derivatives of sin(4x)
Students frequently make mistakes when differentiating sin(4x). 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
Students often forget to apply the chain rule when differentiating composite functions like sin(4x). They might mistakenly write the derivative as cos(4x) instead of 4cos(4x). Ensure that the chain rule is applied correctly by multiplying the derivative of the inner function.
Mistake 2
Forgetting the Factor of 4
Another common mistake is forgetting to multiply by the derivative of the inner function (4x), which is 4. Students should remember that the derivative of sin(kx) is kcos(kx), where k is a constant.
Mistake 3
Incorrect Use of Trigonometric Identities
Students sometimes misapply trigonometric identities, leading to incorrect results. For example, they might incorrectly simplify the derivative of sin(4x) as 4sin(x). Always verify the identities used and ensure they are applied correctly.
Mistake 4
Ignoring the Domain of Trigonometric Functions
Students often ignore the domain of trigonometric functions, leading to errors in calculations. Remember that the cosine function is continuous and defined for all real x.
Mistake 5
Not Simplifying the Equation
Students may forget to simplify the equation, which can lead to incomplete or incorrect results. They often skip steps and directly arrive at the result, especially when solving using the chain rule. Ensure that each step is written in order to avoid errors.
Examples Using the Derivative of sin(4x)
Problem 1
Calculate the derivative of (sin(4x)·cos(4x))
Here, we have f(x) = sin(4x)·cos(4x). Using the product rule, f'(x) = u′v + uv′ In the given equation, u = sin(4x) and v = cos(4x). Let’s differentiate each term, u′= d/dx (sin(4x)) = 4cos(4x) v′= d/dx (cos(4x)) = -4sin(4x) Substituting into the given equation, f'(x) = (4cos(4x))·(cos(4x)) + (sin(4x))·(-4sin(4x)) Let’s simplify terms to get the final answer, f'(x) = 4cos²(4x) - 4sin²(4x) Thus, the derivative of the specified function is 4(cos²(4x) - sin²(4x)).
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 is modeling the growth of a plant using the function y = sin(4x), where y represents the height of the plant at time x. If x = π/8 weeks, calculate the rate of growth of the plant.
We have y = sin(4x) (growth of the plant)...(1) Now, we will differentiate the equation (1) Take the derivative sin(4x): dy/dx = 4cos(4x) Given x = π/8 (substitute this into the derivative) 4cos(4(π/8)) = 4cos(π/2) = 4 × 0 = 0 Hence, the rate of growth of the plant at x = π/8 weeks is 0.
Explanation
We find the rate of growth of the plant at x = π/8 weeks as 0, indicating that at this particular time, the height of the plant is not changing.
Problem 3
Derive the second derivative of the function y = sin(4x).
The first step is to find the first derivative, dy/dx = 4cos(4x)...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [4cos(4x)] = -16sin(4x) Therefore, the second derivative of the function y = sin(4x) is -16sin(4x).
Explanation
We use the step-by-step process, where we start with the first derivative. We then differentiate again, applying the chain rule, to find the second derivative of the function.
Problem 4
Prove: d/dx [(sin(4x))²] = 8sin(4x)cos(4x).
Let’s start using the chain rule: Consider y = (sin(4x))² To differentiate, we use the chain rule: dy/dx = 2sin(4x)·d/dx [sin(4x)] Since the derivative of sin(4x) is 4cos(4x), dy/dx = 2sin(4x)·4cos(4x) = 8sin(4x)cos(4x) Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace sin(4x) with its derivative. As a final step, we simplify to derive the equation.
Problem 5
Solve: d/dx [sin(4x)/x]
To differentiate the function, we use the quotient rule: d/dx [sin(4x)/x] = (d/dx [sin(4x)]·x - sin(4x)·d/dx(x))/x² We will substitute d/dx [sin(4x)] = 4cos(4x) and d/dx(x) = 1 = (4cos(4x)·x - sin(4x)·1)/x² = (4xcos(4x) - sin(4x))/x² Therefore, d/dx [sin(4x)/x] = (4xcos(4x) - sin(4x))/x²
Explanation
In this process, we differentiate the given function using the product rule and quotient rule. As a final step, we simplify the equation to obtain the final result.
FAQs on the Derivative of sin(4x)
1.Find the derivative of sin(4x).
2.Can we use the derivative of sin(4x) in real life?
3.Is it possible to take the derivative of sin(4x) at the point where x = π/4?
4.What rule is used to differentiate sin(4x)/x?
5.Are the derivatives of sin(4x) and sin⁻¹x the same?
Important Glossaries for the Derivative of sin(4x)
- 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 often represented as sin(x), which describes oscillations and periodic phenomena.
- Cosine Function: A trigonometric function that is the derivative of sine and is represented as cos(x).
- Chain Rule: A rule used to differentiate composite functions like sin(kx).
- Quotient Rule: A rule used to differentiate functions that are the quotient of two other functions.
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.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
