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108 LearnersLast updated on October 8, 2025
Derivative of -xy
We use the derivative of -xy to understand how the product of x and y changes in response to a slight change in either variable. Derivatives help us calculate optimal solutions in various real-life situations. We will now discuss the derivative of -xy in detail.
What is the Derivative of -xy?
Let's explore the derivative of -xy. It is commonly represented as d/dx (-xy) or (-xy)', and its value depends on whether x or y is being differentiated.
If x is the variable, the derivative is -y. The function -xy has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below: Product Function: (-xy) Product Rule: Rule for differentiating -xy. Partial Derivatives: Derivatives concerning one variable while holding others constant.
Derivative of -xy Formula
The derivative of -xy can be denoted as d/dx (-xy) or (-xy)'. The formula we use depends on the context: d/dx (-xy) = -y if x is the variable, or d/dy (-xy) = -x if y is the variable.
The formula applies to all x and y within their respective domains.
Proofs of the Derivative of -xy
We can derive the derivative of -xy using various rules. To illustrate, we will use differentiation rules like the product rule.
There are several methods we use to prove this, such as:
- Using the Product Rule
- Using Partial Derivatives
Using the Product Rule
To prove the differentiation of -xy using the product rule: Consider f(x, y) = -xy. Using the product rule: d/dx [u * v] = u' * v + u * v', Let u = -x and v = y. The derivative with respect to x is: d/dx (-xy) = (-1) * y + (-x) * 0 = -y. Thus, d/dx (-xy) = -y.
Using Partial Derivatives
When y is a constant, the derivative of -xy with respect to x is -y. When x is a constant, the derivative of -xy with respect to y is -x.
Higher-Order Derivatives of -xy
When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be complex.
To understand them better, consider a scenario where a change in the product affects the rate of change. Higher-order derivatives provide deeper insights into functions like -xy. For the first derivative of a function, we write f′(x), which indicates how the function changes.
The second derivative is derived from the first derivative, denoted as f′′(x). This pattern continues for higher-order derivatives, denoted as fⁿ(x).
Special Cases
When x or y equals zero, the derivative simplifies significantly as the product becomes zero. When both x and y are constants, the derivative is zero because there is no change.
Common Mistakes and How to Avoid Them in Derivatives of -xy
Students frequently make mistakes when differentiating -xy. These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:
Mistake 1
Ignoring Variable Context
Students may forget to consider whether x or y is the variable of interest, leading to incorrect results.
Always define which variable is being differentiated and apply the derivative accordingly.
Mistake 2
Misapplying the Product Rule
They might not properly apply the product rule to -xy.
Ensure you correctly apply the rule to obtain the correct derivative, noting which variable is held constant.
Mistake 3
Confusion with Partial Derivatives
While differentiating functions like -xy, students may confuse partial derivatives with total derivatives.
Understand that partial derivatives involve holding one variable constant.
Mistake 4
Overlooking Zero Derivatives
Students sometimes overlook cases where the derivative should be zero, such as when the variable is not present in the expression.
Mistake 5
Not Simplifying Results
Students might forget to simplify the derivative expression, leading to unnecessary complexity.
Always simplify the result for clarity.
Examples Using the Derivative of -xy
Problem 1
Calculate the derivative of (-2x * y) with respect to x.
Here, we have f(x, y) = -2x * y. Using the product rule, f'(x) = u′v + uv′. In the given equation, u = -2x and v = y. Let’s differentiate each term: u′ = d/dx (-2x) = -2, v′ = d/dx (y) = 0. Substituting into the given equation, f'(x) = (-2) * y + (-2x) * 0 = -2y. Thus, the derivative of the specified function with respect to x is -2y.
Explanation
We find the derivative of the given function by treating y as a constant and applying the product rule to -2x and y.
Problem 2
A company tracks the interaction between two dependent variables, sales (x) and marketing spend (y), modeled by the function z = -xy. Find the rate of change of z concerning marketing spend when sales are constant at 100 units.
We have z = -xy. Now, differentiate the equation concerning y: dz/dy = -x. Given x = 100, substitute this into the derivative: dz/dy = -100. Hence, the rate of change of z concerning marketing spend is -100 when sales are constant at 100 units.
Explanation
In this scenario, we find the rate of change by treating sales as a constant and differentiating the function -xy concerning marketing spend (y).
Problem 3
Derive the second derivative of the function z = -xy with respect to x.
The first step is to find the first derivative: dz/dx = -y. Now, differentiate the first derivative concerning x to get the second derivative: d²z/dx² = d/dx (-y). As y is treated as a constant with respect to x: d²z/dx² = 0. Therefore, the second derivative of the function z = -xy with respect to x is 0.
Explanation
We start with the first derivative and differentiate it concerning x again, treating y as a constant, resulting in a second derivative of 0.
Problem 4
Prove: d/dy (-x²y) = -x².
Consider z = -x²y. To differentiate with respect to y: dz/dy = -x² * d/dy (y). Since d/dy (y) = 1: dz/dy = -x² * 1 = -x². Hence, proved.
Explanation
In this step-by-step process, we differentiate concerning y, treating x as a constant, to arrive at the derivative -x².
Problem 5
Solve: d/dx (-xy/x).
To differentiate the function, simplify first: z = -y (since -xy/x = -y). The derivative concerning x is: dz/dx = 0, as y is constant. Therefore, d/dx (-xy/x) = 0.
Explanation
In this process, we simplify the expression, noting that it reduces to -y, and then differentiate concerning x, resulting in 0.
FAQs on the Derivative of -xy
1.Find the derivative of -xy concerning x.
2.Can we use the derivative of -xy in real life?
3.Is it possible to take the derivative of -xy when x = 0?
4.What rule is used to differentiate -xy/x?
5.Are the derivatives of -xy and xy the same?
Important Glossaries for the Derivative of -xy
- Derivative: Indicates how a function changes in response to a change in its variables.
- Product Rule: A rule used for differentiating the product of two functions.
- +Partial Derivative: The derivative of a function concerning one variable while holding others constant.
- Constant: A fixed value that does not change.
- Variable: A symbol representing a changeable value in 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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