126 LearnersLast updated on August 5th, 2025
Derivative of 2xy
We use the derivative of 2xy, which helps us understand how the function changes in response to a slight change in x or y. Derivatives are useful in calculating rates of change in real-life situations. We will now discuss the derivative of 2xy in detail.
What is the Derivative of 2xy?
The derivative of 2xy is commonly represented as d/dx (2xy) or (2xy)'.
The function 2xy is differentiable, and its derivative within its domain can be expressed using the product rule.
The key concepts are mentioned below: Product Rule: Rule for differentiating products of functions.
Partial Derivatives: Differentiation concerning one variable while keeping others constant.
Derivative of 2xy Formula
The derivative of 2xy can be denoted as d/dx (2xy) or (2xy)'. When differentiating with respect to x, using the product rule, we get: d/dx (2xy) = 2y + 2x(dy/dx)
The formula applies in scenarios where both x and y can independently vary.
Proofs of the Derivative of 2xy
To derive the derivative of 2xy, we will use the product rule and partial derivatives.
Here's how it's done: Using Product Rule To prove the differentiation of 2xy using the product rule, consider u = 2x and v = y.
Using the product rule: d/dx [u.v] = u'.v + u.v' u' = d/dx (2x) = 2 v' = dy/dx Therefore, d/dx (2xy) = 2y + 2x(dy/dx)
Using Partial Derivatives The derivative of 2xy with respect to x also involves considering partial derivatives.
Treat y as a constant: ∂/∂x (2xy) = 2y For differentiation with respect to y, treat x as a constant: ∂/∂y (2xy) = 2x
Thus, the derivative can be expressed as a combination of these partial derivatives.
Higher-Order Derivatives of 2xy
Higher-order derivatives involve differentiating a function multiple times.
For example, differentiating 2xy with respect to x and then again yields: The first derivative with respect to x is: d/dx (2xy) = 2y + 2x(dy/dx)
The second derivative involves differentiating the first derivative: d²/dx² (2xy) = 2(dy/dx) + 2x(d²y/dx²)
Higher-order derivatives help in understanding the rate of change of the function concerning x and y.
Special Cases:
When y is constant, the derivative reduces to 2y, representing the slope of a line parallel to the x-axis.
When x is constant, the derivative reduces to 2x(dy/dx), indicating the rate of change concerning y.
Common Mistakes and How to Avoid Them in Derivatives of 2xy
Students frequently make mistakes when differentiating 2xy.
These mistakes can be resolved by understanding the proper solutions.
Here are a few common mistakes and ways to solve them:
Mistake 1
Ignoring the Product Rule
Some may apply simple derivative rules without considering the product rule for functions involving products of variables.
Always apply the product rule when differentiating products of variables like 2xy.
Mistake 2
Confusing Partial and Total Derivatives
There is often confusion between partial derivatives and total derivatives.
When differentiating with respect to x, treat y as a constant unless specified otherwise.
Mistake 3
Forgetting to Differentiate Each Variable
In functions involving multiple variables, students may forget to differentiate concerning each variable.
Ensure you apply the derivative to both x and y in expressions like 2xy.
Mistake 4
Incorrect Application of Chain Rule
When functions involve nested derivatives, incorrect application of the chain rule is common.
Break down the function and properly apply chain rule steps for accurate results.
Mistake 5
Overlooking Constants
Students may neglect constants that appear before variables.
Remember to retain constants like 2 in 2xy during differentiation.
Examples Using the Derivative of 2xy
Problem 1
Calculate the derivative of (2xy²).
Here, we have f(x, y) = 2xy².
Using the product rule and treating y² as a function of y, f'(x) = 2y² + 2xy(2y)(dy/dx)
Simplifying, we get: f'(x) = 2y² + 4xy(dy/dx) Thus, the derivative of the specified function is 2y² + 4xy(dy/dx).
Explanation
We find the derivative of the given function by applying the product rule and considering y² as a function of y. This involves differentiating with respect to both x and y.
Problem 2
A rectangular field has an area represented by A = 2xy, where x is the length, and y is the width in meters. If x = 5 meters and y = 3 meters, find the rate of change of area with respect to x.
Given A = 2xy, To find the rate of change concerning x, differentiate A with respect to x: dA/dx = 2y + 2x(dy/dx)
Substitute x = 5, y = 3, and assume dy/dx = 0 (y is constant): dA/dx = 2(3) + 2(5)(0) dA/dx = 6
Hence, the rate of change of the area with respect to x is 6 square meters per meter.
Explanation
We find the rate of change of the area by differentiating the area function with respect to x, treating y as a constant. Substituting the given values provides the final result.
Problem 3
Derive the second derivative of the function f(x, y) = 2xy.
The first step is to find the first derivative, d/dx (2xy) = 2y + 2x(dy/dx)
Now, differentiate the first derivative to get the second derivative: d²/dx² (2xy) = 2(dy/dx) + 2x(d²y/dx²)
Therefore, the second derivative of the function f(x, y) = 2xy is 2(dy/dx) + 2x(d²y/dx²).
Explanation
We use the step-by-step process to find the first derivative and then differentiate it again to obtain the second derivative concerning x.
Problem 4
Prove: d/dx (2x²y) = 4xy + 2x²(dy/dx).
Let's start using the product rule: Consider f(x, y) = 2x²y To differentiate, we use the product rule: df/dx = d/dx (2x²)y + 2x²(dy/dx) = 4xy + 2x²(dy/dx) Hence, proved.
Explanation
In this step-by-step process, we used the product rule to differentiate 2x²y. We broke down the function into parts, differentiated, and then combined the results.
Problem 5
Solve: d/dx (2x/y).
To differentiate the function, we use the quotient rule: d/dx (2x/y) = [d/dx (2x) * y - 2x * d/dx (y)] / y² = [2y - 2x(dy/dx)] / y² Therefore, d/dx (2x/y) = (2y - 2x(dy/dx)) / y²
Explanation
In this process, we differentiate the given function using the quotient rule. We simplify the expression to obtain the final result.
FAQs on the Derivative of 2xy
1.Find the derivative of 2xy.
2.Can the derivative of 2xy be used in real-life applications?
3.Is it possible to take the derivative of 2xy when y is a function of x?
4.What rule is used to differentiate 2x/y?
5.Are the derivatives of 2xy and 2yx the same?
Important Glossaries for the Derivative of 2xy
- Derivative: The derivative of a function indicates how the function changes concerning a slight change in variables.
- Product Rule: A rule used to differentiate the product of two functions.
- Partial Derivative: The derivative of a function of multiple variables with respect to one variable, keeping others constant.
- Quotient Rule: A rule used to differentiate the ratio of two functions.
- Second Derivative: The derivative of the first derivative, indicating the change in the rate of change of a function.
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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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