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Last updated on July 21st, 2025

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Derivative of 2xy

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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.

Derivative of 2xy for Thai Students
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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.

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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.

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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.

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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.

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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.

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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

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Ignoring the Product Rule

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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

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Confusing Partial and Total Derivatives

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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

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Forgetting to Differentiate Each Variable

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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

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Incorrect Application of Chain Rule

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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

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Overlooking Constants

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Students may neglect constants that appear before variables.

 

Remember to retain constants like 2 in 2xy during differentiation.

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Examples Using the Derivative of 2xy

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Problem 1

Calculate the derivative of (2xy²).

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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.

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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.

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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.

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Problem 3

Derive the second derivative of the function f(x, y) = 2xy.

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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.

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Problem 4

Prove: d/dx (2x²y) = 4xy + 2x²(dy/dx).

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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.

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Problem 5

Solve: d/dx (2x/y).

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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.

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FAQs on the Derivative of 2xy

1.Find the derivative of 2xy.

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2.Can the derivative of 2xy be used in real-life applications?

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3.Is it possible to take the derivative of 2xy when y is a function of x?

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4.What rule is used to differentiate 2x/y?

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5.Are the derivatives of 2xy and 2yx the same?

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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

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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Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

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