103 LearnersLast updated on July 16th, 2025
Derivative of 3cosx
We use the derivative of 3cos(x), which is -3sin(x), to understand 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 discuss the derivative of 3cos(x) in detail.
What is the Derivative of 3cosx?
We now understand the derivative of 3cos(x). It is commonly represented as d/dx (3cos x) or (3cos x)', and its value is -3sin(x). The function 3cos(x) has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Cosine Function: cos(x)
Constant Multiple Rule: Rule for differentiating a constant times a function.
Sine Function: sin(x)
Derivative of 3cosx Formula
The derivative of 3cos(x) can be denoted as d/dx (3cos x) or (3cos x)'.
The formula we use to differentiate 3cos(x) is: d/dx (3cos x) = -3sin x
The formula applies to all x where the function is defined.
Proofs of the Derivative of 3cosx
We can derive the derivative of 3cos(x) 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:
- Using Chain Rule
- Using Constant Multiple Rule
We will now demonstrate that the differentiation of 3cos(x) results in -3sin(x) using these methods:
- Using Chain Rule To prove the differentiation of 3cos(x) using the chain rule, Consider f(x) = 3cos(x)
We use the formula: d/dx [a·f(x)] = a·f'(x) Let a = 3 and f(x) = cos(x) f'(x) = d/dx (cos x) = -sin x Thus, d/dx (3cos x) = 3·(-sin x) = -3sin x
- Using Constant Multiple Rule The derivative of a constant times a function is the constant times the derivative of the function. Given y = 3cos(x), dy/dx = 3·d/dx (cos x) = 3·(-sin x) = -3sin x Hence, proved.
Higher-Order Derivatives of 3cosx
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 3cos(x).
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 3cos(x), 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 π/2, the derivative is -3sin(π/2), which is -3. When x is 0, the derivative of 3cos x = -3sin(0), which is 0.
Common Mistakes and How to Avoid Them in Derivatives of 3cosx
Students frequently make mistakes when differentiating 3cos(x). These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:
Mistake 1
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 product or chain rule. Ensure that each step is written in order. Students might think it is awkward, but it is important to avoid errors in the process.
Mistake 2
Forgetting the sign change of sin(x)
They might not remember that the derivative of cos(x) is -sin(x), leading to sign errors. It is crucial to remember this sign change to avoid mistakes in calculations.
Mistake 3
Incorrect use of Constant Multiple Rule
While differentiating functions such as 3cos(x), students might forget to multiply the constant correctly. For example, incorrect differentiation: d/dx (3cos x) = 3cos x · -sin x. The correct approach is: d/dx (3cos x) = 3·d/dx (cos x) = 3·(-sin x) = -3sin x. To avoid this mistake, apply the constant multiple rule properly.
Mistake 4
Not writing Constants and Coefficients
There is a common mistake that students at times forget to multiply the constants placed before cos(x). For example, they incorrectly write d/dx (5cos x) = -sin x. Students should check the constants in the terms and ensure they are multiplied properly. For instance, the correct equation is d/dx (5cos x) = -5sin x.
Mistake 5
Misapplying Trigonometric Identities
Students often forget basic trigonometric identities which can lead to mistakes. For example, forgetting that sin²(x) + cos²(x) = 1. These identities are useful in simplifying derivatives and avoiding errors.
Examples Using the Derivative of 3cosx
Problem 1
Calculate the derivative of 3cos(x)·e^x
Here, we have f(x) = 3cos(x)·e^x.
Using the product rule, f'(x) = u′v + uv′ In the given equation, u = 3cos(x) and v = e^x.
Let’s differentiate each term, u′ = d/dx (3cos x) = -3sin x v′ = d/dx (e^x) = e^x Substituting into the given equation, f'(x) = (-3sin x)·(e^x) + (3cos x)·(e^x)
Let’s simplify terms to get the final answer, f'(x) = e^x(-3sin x + 3cos x)
Thus, the derivative of the specified function is e^x(-3sin x + 3cos x).
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
The height of a Ferris wheel is represented by the function y = 3cos(x), where y is the height of a specific point at an angle x. If x = π/3 radians, determine the rate at which the height changes.
We have y = 3cos(x) (height of the Ferris wheel)...(1)
Now, we will differentiate the equation (1). Take the derivative of 3cos(x): dy/dx = -3sin(x) Given x = π/3 (substitute this into the derivative)
dy/dx = -3sin(π/3) = -3(√3/2) = -3√3/2
Hence, the rate of change of height at x = π/3 is -3√3/2.
Explanation
We find the rate of change of height at x = π/3 by differentiating the height function. This shows how quickly the height of the Ferris wheel changes at a certain angle.
Problem 3
Derive the second derivative of the function y = 3cos(x).
The first step is to find the first derivative, dy/dx = -3sin(x)...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [-3sin(x)] d²y/dx² = -3cos(x)
Therefore, the second derivative of the function y = 3cos(x) is -3cos(x).
Explanation
We use the step-by-step process, where we start with the first derivative. Then we differentiate -3sin(x) to find the second derivative, resulting in -3cos(x).
Problem 4
Prove: d/dx (3cos²x) = -6cos(x)sin(x).
Let’s start using the chain rule: Consider y = 3cos²(x) = 3[cos(x)]²
To differentiate, we use the chain rule: dy/dx = 3·2cos(x)·d/dx [cos(x)]
Since the derivative of cos(x) is -sin(x), dy/dx = 6cos(x)(-sin(x)) = -6cos(x)sin(x)
Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace cos(x) with its derivative. As a final step, we simplify to derive the equation.
Problem 5
Solve: d/dx (3cos x/x)
To differentiate the function, we use the quotient rule: d/dx (3cos x/x) = (d/dx (3cos x)·x - 3cos x·d/dx(x))/x²
We will substitute d/dx (3cos x) = -3sin x and d/dx (x) = 1 = (-3sin x·x - 3cos x·1)/x² = (-3xsin x - 3cos x)/x²
Therefore, d/dx (3cos x/x) = (-3xsin x - 3cos x)/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.
FAQs on the Derivative of 3cosx
1.Find the derivative of 3cos(x).
2.Can we use the derivative of 3cos(x) in real life?
3.What happens to the derivative of 3cos(x) at x = π/2?
4.What rule is used to differentiate 3cos(x)/x?
5.How does the derivative of 3cos(x) relate to the derivative of cos(x)?
Important Glossaries for the Derivative of 3cosx
- 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 representing the ratio of the adjacent side to the hypotenuse.
- Sine Function: A trigonometric function representing the ratio of the opposite side to the hypotenuse.
- Constant Multiple Rule: A rule stating that the derivative of a constant times a function is the constant times the derivative of the function.
- Chain Rule: A formula for computing the derivative of the composition of two or more 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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