Derivative of xy
2026-02-28 10:26 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-xy

589 Learners

Last updated on August 5, 2025

We use the derivative of xy, which is y + x(dy/dx), as a measuring tool for how the product of two variables changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now talk about the derivative of xy in detail.

What is the Derivative of xy?

We now understand the derivative of xy. It is commonly represented as d/dx (xy) or (xy)', and its value is y + x(dy/dx). The function xy has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Product Function: (xy = x · y). Product Rule: Rule for differentiating xy (since it consists of the product of two functions).

Derivative of xy Formula

The derivative of xy can be denoted as d/dx (xy) or (xy)'. The formula we use to differentiate xy is: d/dx (xy) = y + x(dy/dx) The formula applies to all x where both x and y are differentiable functions of x.

Proofs of the Derivative of xy

We can derive the derivative of xy using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as: Using Product Rule We will now demonstrate that the differentiation of xy results in y + x(dy/dx) using the above-mentioned method: Using Product Rule To prove the differentiation of xy using the product rule, We use the formula: d/dx [u · v] = u' · v + u · v' Consider u = x and v = y So we get, xy = u · v By product rule: d/dx (xy) = (dx/dx) · y + x · (dy/dx) d/dx (xy) = y + x(dy/dx) Hence, proved.

Explore Our Programs

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 a little tricky. To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like xy. For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(x) Similarly, the third derivative, f′′′(x) is the result of the second derivative, and this pattern continues. For the nth Derivative of xy, we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change (continuing for higher-order derivatives).

Special Cases:

When y is a constant, the derivative simplifies to x(dy/dx). When x is a constant, the derivative simplifies to y.

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:

Problem 1

Calculate the derivative of (3xy)

Okay, lets begin

Here, we have f(x) = 3xy. Using the product rule, f'(x) = 3(y + x(dy/dx)) In the given equation, we have a constant multiplier 3. Thus, the derivative of the specified function is 3(y + x(dy/dx)).

Explanation

We find the derivative of the given function by applying the product rule and considering the constant multiplier.

Well explained 👍

Problem 2

A company uses the formula C = xy to calculate the cost of production, where x is the number of units and y is the cost per unit. If x = 100 units and the cost per unit changes at a rate of 2 per unit, find the rate of change of the total cost.

Okay, lets begin

We have C = xy (total cost)...(1) Differentiate equation (1) with respect to x: dC/dx = y + x(dy/dx) Given x = 100 and dy/dx = 2, substitute these into the derivative: dC/dx = y + 100 * 2 dC/dx = y + 200 Hence, the rate of change of the total cost is y + 200.

Explanation

We find the rate of change of total cost by differentiating the cost function with respect to x and substituting the given values.

Well explained 👍

Problem 3

Derive the second derivative of the function y = xy.

Okay, lets begin

The first step is to find the first derivative, dy/dx = y + x(dy/dx)...(1) Now differentiate equation (1) to get the second derivative: d²y/dx² = dy/dx + (dy/dx + x(d²y/dx²)) Simplify terms to get the final answer: d²y/dx² = 2(dy/dx) + x(d²y/dx²).

Explanation

We use the step-by-step process, where we start with the first derivative. We then differentiate again to find the second derivative.

Well explained 👍

Problem 4

Prove: d/dx (x²y) = 2xy + x²(dy/dx).

Okay, lets begin

Let’s start using the product rule: Consider u = x² and v = y d/dx (x²y) = (d/dx (x²)) · y + x² · (dy/dx) d/dx (x²y) = 2x · y + x² · (dy/dx) d/dx (x²y) = 2xy + x²(dy/dx) Hence proved.

Explanation

In this step-by-step process, we used the product rule to differentiate the equation, showing each step clearly.

Well explained 👍

Problem 5

Solve: d/dx (xy²)

Okay, lets begin

To differentiate the function, we use the product rule: d/dx (xy²) = (d/dx (x)) · y² + x · d/dx (y²) = y² + x · (2y(dy/dx)) = y² + 2xy(dy/dx) Therefore, d/dx (xy²) = y² + 2xy(dy/dx).

Explanation

In this process, we differentiate the given function using the product rule, applying it to both parts of the function.

Well explained 👍

FAQs on the Derivative of xy

1.Find the derivative of xy.

Using the product rule for xy gives: d/dx (xy) = y + x(dy/dx).

2.Can we use the derivative of xy in real life?

Yes, we can use the derivative of xy in real life in calculating the rate of change of related variables, especially in fields such as physics, economics, and engineering.

3.Is it possible to take the derivative of xy if y is a constant?

Yes, if y is a constant, the derivative simplifies to x(dy/dx), which is zero since dy/dx = 0.

4.What rule is used to differentiate xy?

We use the product rule to differentiate xy: d/dx (xy) = y + x(dy/dx).

5.Are the derivatives of xy and x²y the same?

No, they are different. The derivative of xy is y + x(dy/dx), while the derivative of x²y is 2xy + x²(dy/dx).

Important Glossaries for the Derivative of xy

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Product Rule: A differentiation rule used to differentiate functions that are the product of two other functions. First Derivative: The initial result of a function, which gives us the rate of change of a specific function. Higher-Order Derivative: Derivatives of a function taken multiple times, providing insights into the changing rates of change. Constant: A value that does not change and has no derivative with respect to a variable.

What Is Calculus? 🔢 | Easy Tricks, Limits & 🎯 Fun Learning for Kids | ✨BrightCHAMPS Math

▶

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.

Fun Fact

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