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108 LearnersLast updated on September 27, 2025
Derivative of 4xy
We use the derivative of 4xy to understand how this function changes in response to a slight change in x and y. Derivatives help us calculate profit or loss in real-life situations. We will now talk about the derivative of 4xy in detail.
What is the Derivative of 4xy?
We now understand the derivative of 4xy. It is commonly represented as d/dx (4xy) or (4xy)', and its value depends on the context in which it is differentiated. The function 4xy has a clearly defined derivative, indicating it is differentiable with respect to x or y.
The key concepts are mentioned below:
Product Rule: Rule for differentiating products of functions, like 4xy.
Partial Derivatives: When dealing with functions of multiple variables, we differentiate with respect to one variable while keeping others constant.
Derivative of 4xy Formula
The derivative of 4xy can be expressed using partial derivatives or the product rule. If you're differentiating with respect to x, then: ∂/∂x (4xy) = 4y
If you're differentiating with respect to y, then: ∂/∂y (4xy) = 4x These formulas apply to all x and y in the domain of the function.
Proofs of the Derivative of 4xy
We can derive the derivative of 4xy using proofs. To show this, we will use the product rule and partial derivatives.
There are several methods we use to prove this, such as:
Using Product Rule
For the function 4xy, we apply the product rule: To differentiate with respect to x: d/dx (4xy) = 4 * d/dx (xy) = 4 * (y + x * dy/dx) If we assume y is a constant, then dy/dx = 0: d/dx (4xy) = 4y To differentiate with respect to y: d/dy (4xy) = 4 * d/dy (xy) = 4 * (x + y * dx/dy) If we assume x is a constant, then dx/dy = 0: d/dy (4xy) = 4x
Using Partial Derivatives
When using partial derivatives, we treat other variables as constants: ∂/∂x (4xy) = 4y ∂/∂y (4xy) = 4x Hence, the derivative of 4xy can be calculated using either method.
Higher-Order Derivatives of 4xy
When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. For a simple product like 4xy, higher-order derivatives involve further differentiation with respect to x or y. To understand them better, consider how further differentiation affects each variable.
For the first derivative with respect to x, we write f′(x) = 4y, which indicates how the function changes with a change in x. For the second derivative with respect to x, f′′(x) = 0, indicating no further change with respect to x alone.
For higher-order derivatives, this pattern continues, leading to zero since each differentiation removes the variable.
Special Cases:
When either x or y is zero, the derivative with respect to the other variable becomes zero as the entire term 4xy becomes zero.
When x or y is a constant, the derivative with respect to the other variable is simply the constant multiplied by 4.
Common Mistakes and How to Avoid Them in Derivatives of 4xy
Students frequently make mistakes when differentiating 4xy. These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:
Mistake 1
Not using the Product Rule correctly
Students may forget to apply the product rule correctly, especially when differentiating with respect to one variable and keeping the other constant.
Ensure that each step is written in order, applying the product rule when necessary.
Mistake 2
Forgetting to treat variables as constants
When using partial derivatives, it's crucial to treat all other variables as constants.
Students might incorrectly differentiate both variables simultaneously instead of focusing on one at a time.
Mistake 3
Mixing up partial derivatives
While differentiating, students sometimes confuse the partial derivative symbols or forget which variable they are differentiating with respect to.
Always double-check which variable you are focusing on and ensure the correct notation.
Mistake 4
Not simplifying derivatives correctly
Simplifying derivatives can lead to mistakes if not done correctly.
Make sure to simplify each term properly and keep track of constants and coefficients.
Mistake 5
Ignoring higher-order derivatives
Students may overlook the effects of higher-order derivatives.
Understand that for simple functions like 4xy, higher derivatives may result in zero or constant terms, but should still be calculated correctly.
Examples Using the Derivative of 4xy
Problem 1
Calculate the derivative of 4xy² with respect to x.
Here, we have f(x, y) = 4xy². Using partial derivatives, we treat y as a constant: ∂/∂x (4xy²) = 4y² Thus, the derivative of the specified function with respect to x is 4y².
Explanation
We find the derivative of the given function by treating y as a constant and differentiating with respect to x to obtain the final result.
Problem 2
A company produces widgets at a rate modeled by w(x, y) = 4xy, where x is the number of workers and y is the hours worked. If x = 10 workers, find the rate of change of production with respect to hours worked.
We have w(x, y) = 4xy. To find the rate of change with respect to y, differentiate partially with respect to y: ∂w/∂y = 4x Substitute x = 10: ∂w/∂y = 4(10) = 40 Hence, the rate of change of production with respect to hours worked is 40 widgets per hour.
Explanation
We find the rate of change of production by treating x as a constant and differentiating with respect to y.
Substituting the given value of x gives us the final rate.
Problem 3
Derive the second derivative of the function 4xy² with respect to x.
The first step is to find the first derivative with respect to x: ∂/∂x (4xy²) = 4y² Now, differentiate again with respect to x to get the second derivative: ∂²/∂x² (4xy²) = 0 Therefore, the second derivative of the function 4xy² with respect to x is 0.
Explanation
We use a step-by-step process where we start with the first derivative and differentiate again with respect to x, resulting in zero for the second derivative.
Problem 4
Prove: ∂/∂x (4x²y) = 8xy.
Let’s start using partial derivatives: Consider the function 4x²y. Differentiate with respect to x: ∂/∂x (4x²y) = 4 * 2xy = 8xy Hence, proved.
Explanation
In this step-by-step process, we use partial derivatives to differentiate the equation with respect to x and simplify the result to derive the equation.
Problem 5
Solve: ∂/∂y (4xy/2).
To differentiate the function, treat x as a constant: ∂/∂y (4xy/2) = (4x/2) = 2x Therefore, ∂/∂y (4xy/2) = 2x.
Explanation
In this process, we differentiate the given function with respect to y while treating x as a constant and simplify the equation to obtain the final result.
FAQs on the Derivative of 4xy
1.Find the derivative of 4xy with respect to x.
2.Can we use the derivative of 4xy in real life?
3.Is it possible to take the derivative of 4xy at any point?
4.What rule is used to differentiate 4xy?
5.Are the derivatives of 4xy and (4xy)² the same?
Important Glossaries for the Derivative of 4xy
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in one of its variables.
- Product Rule: A differentiation rule used to find the derivative of the product of two functions.
- Partial Derivative: The derivative of a function with multiple variables with respect to one variable while keeping others constant.
- Higher-Order Derivative: Derivatives obtained by differentiating a function multiple times.
- Constant: A fixed value that does not change in the context of differentiation. ```
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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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: He loves to play the quiz with kids through algebra to make kids love it.
