114 LearnersLast updated on August 5th, 2025
Derivative of cos(x^3)
We use the derivative of cos(x^3), which involves applying the chain rule, as a tool to measure how the 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 cos(x^3) in detail.
What is the Derivative of cos(x^3)?
We now understand the derivative of cos(x^3). It is commonly represented as d/dx (cos(x^3)) or (cos(x^3))', and its value is -3x²sin(x^3).
The function cos(x^3) has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below:
Cosine Function: cos(x^3) involves the composition of functions.
Chain Rule: Rule for differentiating cos(x^3) due to its composite nature.
Sine Function: sin(x) is the derivative of cos(x).
Derivative of cos(x^3) Formula
The derivative of cos(x^3) can be denoted as d/dx (cos(x^3)) or (cos(x^3))'. The formula we use to differentiate cos(x^3) is: d/dx (cos(x^3)) = -3x²sin(x^3)
formula applies to all x where x is within the domain of the function.
Proofs of the Derivative of cos(x^3)
We can derive the derivative of cos(x^3) 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 aBy Chain Rule
We will now demonstrate that the differentiation of cos(x^3) results in -3x²sin(x^3) using the chain rule:
Using Chain Rule
To prove the differentiation of cos(x^3) using the chain rule, We use the formula: cos(x^3) = cos(u), where u = x^3
The derivative of cos(u) is -sin(u), and the derivative of u = x^3 is 3x².
By chain rule: d/dx [cos(u)] = -sin(u) * du/dx
Let’s substitute u = x^3, d/dx (cos(x^3)) = -sin(x^3) * 3x²
Hence, d/dx (cos(x^3)) = -3x²sin(x^3).
Higher-Order Derivatives of cos(x^3)
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(x^3).
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(x^3), 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=0, the derivative of cos(x^3) = -3x²sin(x^3) = 0 because sin(0) = 0.
When x is not a real number, the derivative is undefined because x^3 must be a real number for cos(x^3) to be defined.
Common Mistakes and How to Avoid Them in Derivatives of cos(x^3)
Students frequently make mistakes when differentiating cos(x^3). 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 correctly, which can lead to incomplete or incorrect results. They often skip steps and directly arrive at the result. Ensure that each step is written in order. It's important to avoid errors in the process by carefully differentiating the inner function as well.
Mistake 2
Ignoring the power of x in x^3
Students might not remember to include the derivative of x^3, which is 3x². This is crucial for applying the chain rule. Always remember to differentiate the inner function as well.
Mistake 3
Incorrect simplification of trigonometric expressions
While differentiating functions like cos(x^3), students may misapply trigonometric identities. For example, incorrectly simplifying expressions involving sin(x) and cos(x). Always verify trigonometric simplifications and ensure they are properly applied.
Mistake 4
Forgetting to multiply by the derivative of the inner function
There is a common mistake that students sometimes forget to multiply by the derivative of the inner function x^3. This can lead to incorrect results. Always check each term for the presence of the inner derivative.
Mistake 5
Misidentifying the derivative of cosine
Students often forget that the derivative of cos(x) is -sin(x). This happens when they rush through calculations. Ensure you remember this fundamental derivative rule to avoid mistakes.
Examples Using the Derivative of cos(x^3)
Problem 1
Calculate the derivative of cos(x^3)·x²
Here, we have f(x) = cos(x^3)·x². Using the product rule, f'(x) = u′v + uv′ In the given equation, u = cos(x^3) and v = x².
Let’s differentiate each term, u′ = d/dx (cos(x^3)) = -3x²sin(x^3) v′ = d/dx (x²) = 2x
Substituting into the given equation, f'(x) = (-3x²sin(x^3))·x² + cos(x^3)·2x
Let’s simplify terms to get the final answer, f'(x) = -3x⁴sin(x^3) + 2xcos(x^3)
Thus, the derivative of the specified function is -3x⁴sin(x^3) + 2xcos(x^3).
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 analyzing the cost function represented by y = cos(x^3), where y represents the cost at a production level x. If x = 1 meter, determine the rate of change of the cost.
We have y = cos(x^3) (cost function)...(1)
Now, we will differentiate the equation (1)
Take the derivative of cos(x^3): dy/dx = -3x²sin(x^3)
Given x = 1 (substitute this into the derivative) dy/dx = -3(1)²sin(1^3) dy/dx = -3sin(1)
Hence, the rate of change of the cost at x = 1 is -3sin(1).
Explanation
We find the rate of change of the cost at x = 1, which provides insight into how the cost function behaves at that particular production level.
Problem 3
Derive the second derivative of the function y = cos(x^3).
The first step is to find the first derivative, dy/dx = -3x²sin(x^3)...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [-3x²sin(x^3)]
Here we use the product rule, d²y/dx² = -3[d/dx(x²)sin(x^3) + x²d/dx(sin(x^3))] d²y/dx² = -3[2xsin(x^3) + x²(3x²cos(x^3))] d²y/dx² = -3[2xsin(x^3) + 3x⁴cos(x^3)]
Therefore, the second derivative of the function y = cos(x^3) is -3(2xsin(x^3) + 3x⁴cos(x^3)).
Explanation
We use the step-by-step process, starting with the first derivative. Using the product rule, we differentiate the components and simplify the terms to find the final answer.
Problem 4
Prove: d/dx (sin(x^3)) = 3x²cos(x^3).
Let’s start using the chain rule: Consider y = sin(x^3)
The derivative of sin(u) is cos(u), and the derivative of u = x^3 is 3x².
Using the chain rule: dy/dx = cos(x^3) * 3x²
Hence, d/dx (sin(x^3)) = 3x²cos(x^3).
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. We replace sin(x^3) with its derivative and multiply by the derivative of the inner function to derive the equation.
Problem 5
Solve: d/dx (cos(x^3)/x)
To differentiate the function, we use the quotient rule: d/dx (cos(x^3)/x) = (d/dx (cos(x^3))·x - cos(x^3)·d/dx(x))/x²
We will substitute d/dx (cos(x^3)) = -3x²sin(x^3) and d/dx(x) = 1 = (-3x²sin(x^3)·x - cos(x^3)·1)/x² = (-3x³sin(x^3) - cos(x^3))/x²
Therefore, d/dx (cos(x^3)/x) = (-3x³sin(x^3) - cos(x^3))/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 cos(x^3)
1.Find the derivative of cos(x^3).
2.Can we use the derivative of cos(x^3) in real life?
3.Is it possible to take the derivative of cos(x^3) at x = 0?
4.What rule is used to differentiate cos(x^3)/x?
5.Are the derivatives of cos(x^3) and cos⁻¹(x³) the same?
Important Glossaries for the Derivative of cos(x^3)
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Chain Rule: A fundamental rule in calculus used for differentiating composite functions.
- Cosine Function: A trigonometric function, written as cos(x), which describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
- Sine Function: A trigonometric function that is the derivative of the cosine function.
- Quotient Rule: A method for differentiating functions that are divided by one another.
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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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