Last updated on July 21st, 2025
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.
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.
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.
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 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.
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.
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:
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).
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.
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.
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.
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).
We use the step-by-step process to find the first derivative and then differentiate it again to obtain the second derivative concerning x.
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.
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.
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
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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