103 LearnersLast updated on 6 September 2025
Derivative of Cos(3x)
We use the derivative of cos(3x), which is -3sin(3x), as a measuring tool for how the cosine function changes in response to a slight change in x. Derivatives help us calculate various changes in real-life situations. We will now talk about the derivative of cos(3x) in detail.
What is the Derivative of Cos(3x)?
We now understand the derivative of cos(3x).
It is commonly represented as d/dx (cos(3x)) or (cos(3x))', and its value is -3sin(3x).
The function cos(3x) has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Cosine Function: (cos(x) = adjacent/hypotenuse).
Chain Rule: Rule for differentiating cos(3x) (since it involves a function within another function).
Sine Function: sin(x) is a trigonometric function.
Derivative of Cos(3x) Formula
The derivative of cos(3x) can be denoted as d/dx (cos(3x)) or (cos(3x))'.
The formula we use to differentiate cos(3x) is: d/dx (cos(3x)) = -3sin(3x)
The formula applies to all x where the derivative is defined.
Proofs of the Derivative of Cos(3x)
We can derive the derivative of cos(3x) 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
We will now demonstrate that the differentiation of cos(3x) results in -3sin(3x) using the above-mentioned methods:
By First Principle
The derivative of cos(3x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.
To find the derivative of cos(3x) using the first principle, we will consider f(x) = cos(3x).
Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)
Given that f(x) = cos(3x), we write f(x + h) = cos(3(x + h)).
Substituting these into equation (1), f'(x) = limₕ→₀ [cos(3(x + h)) - cos(3x)] / h = limₕ→₀ [-2sin(3x + 3h/2)sin(3h/2)] / h
Using the limit formula, limₕ→₀ sin(h)/h = 1, we have: f'(x) = -3sin(3x)
Hence, proved.
Using Chain Rule
To prove the differentiation of cos(3x) using the chain rule, We use the formula:
Let u = 3x, so that cos(3x) = cos(u)
By chain rule: d/dx [cos(u)] = -sin(u) * d/dx[3x] d/dx (cos(3x)) = -sin(3x) * 3 = -3sin(3x)
Higher-Order Derivatives of Cos(3x)
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 cos(3x).
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 cos(3x), 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 such that 3x = π/2, the derivative is -3sin(π/2), which is -3. When x = 0, the derivative of cos(3x) = -3sin(0), which is 0.
Common Mistakes and How to Avoid Them in Derivatives of Cos(3x)
Students frequently make mistakes when differentiating cos(3x). 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, leading to incorrect results. They often skip the multiplication by the derivative of the inner function, especially when differentiating composite functions like cos(3x). Ensure that each step is written in order and don't forget to multiply by the derivative of 3x, which is 3.
Mistake 2
Forgetting the negative sign
They might not remember that the derivative of cosine involves a negative sign. Keep in mind that the derivative of cos(x) is -sin(x), and thus the derivative of cos(3x) is -3sin(3x).
Mistake 3
Incorrect use of Trigonometric Identities
While differentiating functions involving trigonometric identities, students might misapply identities.
For example: Incorrect differentiation: d/dx (cos^2(3x)) = 2cos(3x)(-3sin(3x)) instead of using proper identities.
To avoid this mistake, always check for the appropriate trigonometric identity and apply it correctly.
Mistake 4
Not writing Constants and Coefficients
There is a common mistake that students at times forget to multiply the constants placed before cos(3x).
For example, they incorrectly write d/dx (5 cos(3x)) = -3sin(3x).
Students should check the constants in the terms and ensure they are multiplied properly.
For e.g., the correct equation is d/dx (5 cos(3x)) = -15sin(3x).
Mistake 5
Misinterpreting the Scope of Differentiation
Students often forget to differentiate the entire argument of the trigonometric function.
For example: Incorrect: d/dx (cos(3x + 2)) = -3sin(3x + 2). To fix this error, students should differentiate the entire argument and apply the chain rule correctly.
Examples Using the Derivative of Cos(3x)
Problem 1
Calculate the derivative of (cos(3x)·e^x)
Here, we have f(x) = cos(3x)·ex.
Using the product rule, f'(x) = u′v + uv′ In the given equation, u = cos(3x) and v = ex.
Let’s differentiate each term, u′= d/dx (cos(3x)) = -3sin(3x) v′= d/dx (ex) = ex substituting into the given equation, f'(x) = (-3sin(3x)).(ex) + (cos(3x)).(ex)
Let’s simplify terms to get the final answer, f'(x) = ex(-3sin(3x) + cos(3x))
Thus, the derivative of the specified function is ex(-3sin(3x) + cos(3x)).
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 monitors the oscillation of a spring, represented by the function y = cos(3x), where y represents the displacement of the spring at time x. If x = π/6 seconds, calculate the rate of change of displacement.
We have y = cos(3x) (displacement function)...(1)
Now, we will differentiate the equation (1) Take the derivative cos(3x): dy/dx = -3sin(3x)
Given x = π/6 (substitute this into the derivative) dy/dx = -3sin(3(π/6)) dy/dx = -3sin(π/2) dy/dx = -3(1) = -3
Hence, we get the rate of change of displacement at time x = π/6 as -3.
Explanation
We find the rate of change of displacement at x = π/6 as -3, which means that at this point, the displacement of the spring is decreasing at a rate of 3 units per second.
Problem 3
Derive the second derivative of the function y = cos(3x).
The first step is to find the first derivative, dy/dx = -3sin(3x)...(1)
Now we will differentiate equation (1) to get the second derivative:
d²y/dx² = d/dx [-3sin(3x)]
Here we use the chain rule, d²y/dx² = -3cos(3x) * 3 d²y/dx² = -9cos(3x)
Therefore, the second derivative of the function y = cos(3x) is -9cos(3x).
Explanation
We use the step-by-step process, where we start with the first derivative. Using the chain rule, we differentiate -3sin(3x). We then simplify the terms to find the final answer.
Problem 4
Prove: d/dx (cos²(3x)) = -6cos(3x)sin(3x).
Let’s start using the chain rule: Consider y = cos²(3x) [cos(3x)]²
To differentiate, we use the chain rule: dy/dx = 2cos(3x) * d/dx [cos(3x)]
Since the derivative of cos(3x) is -3sin(3x), dy/dx = 2cos(3x) * (-3sin(3x)) dy/dx = -6cos(3x)sin(3x)
Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation.
Then, we replace cos(3x) with its derivative.
As a final step, we substitute y = cos²(3x) to derive the equation.
Problem 5
Solve: d/dx (cos(3x)/x)
To differentiate the function, we use the quotient rule: d/dx (cos(3x)/x) = (d/dx (cos(3x)).x - cos(3x).d/dx(x)) / x²
We will substitute d/dx (cos(3x)) = -3sin(3x) and d/dx (x) = 1 = (-3sin(3x).x - cos(3x)) / x² = (-3xsin(3x) - cos(3x)) / x²
Therefore, d/dx (cos(3x)/x) = (-3xsin(3x) - cos(3x)) / 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 Cos(3x)
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⁻¹(3x) the same?
6.Can we find the derivative of the cos(3x) formula?
Important Glossaries for the Derivative of Cos(3x)
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Cosine Function: A trigonometric function that represents the adjacent over the hypotenuse in a right triangle.
- Sine Function: A trigonometric function representing the opposite over the hypotenuse.
- Chain Rule: A fundamental rule in calculus used to differentiate the composition of functions.
- Quotient Rule: A rule for differentiating functions that are divided by each other.
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.
