Derivative of xe^x
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Last updated on August 5, 2025

We use the derivative of xe^x, which is (x + 1)e^x, as a tool to understand how the function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now discuss the derivative of xe^x in detail.

What is the Derivative of xe^x?

We now understand the derivative of xe^x. It is commonly represented as d/dx (xe^x) or (xe^x)', and its value is (x + 1)e^x. The function xe^x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Exponential Function: (e^x is a fundamental mathematical constant raised to the power of x). Product Rule: Rule for differentiating xe^x (since it is a product of x and e^x). Constant Rule: The derivative of a constant is zero.

Derivative of xe^x Formula

The derivative of xe^x can be denoted as d/dx (xe^x) or (xe^x)'. The formula we use to differentiate xe^x is: d/dx (xe^x) = (x + 1)e^x (xe^x)' = (x + 1)e^x This formula applies to all x in the domain of real numbers.

Proofs of the Derivative of xe^x

We can derive the derivative of xe^x using proofs. To show this, we will use differentiation rules. There are several methods we use to prove this, such as: Using Product Rule Using First Principle Using Product Rule To prove the differentiation of xe^x using the product rule, We use the formula: xe^x = x(e^x) The product rule states: d/dx [u·v] = u'·v + u·v' Let u = x and v = e^x u' = d/dx (x) = 1 v' = d/dx (e^x) = e^x Substituting into the product rule formula: d/dx (xe^x) = 1·e^x + x·e^x = e^x + xe^x = (x + 1)e^x Using First Principle The derivative of xe^x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of xe^x using the first principle, we will consider f(x) = xe^x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given that f(x) = xe^x, we write f(x + h) = (x + h)e^(x + h). Substituting these into the equation, f'(x) = limₕ→₀ [(x + h)e^(x + h) - xe^x] / h = limₕ→₀ [xe^(x + h) + he^(x + h) - xe^x] / h = limₕ→₀ [xe^(x + h) - xe^x + he^(x + h)] / h = limₕ→₀ [e^x(x + h - x) / h + he^x] = limₕ→₀ [e^x + he^x/h] = e^x + e^x = (x + 1)e^x Hence, proved.

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Higher-Order Derivatives of xe^x

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 xe^x. 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 xe^x, 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 x is 0, the derivative of xe^x = (0 + 1)e^0, which is 1. When x tends to negative infinity, the derivative approaches 0 as e^x becomes very small.

Common Mistakes and How to Avoid Them in Derivatives of xe^x

Students frequently make mistakes when differentiating xe^x. 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 (xe^x·e^x)

Okay, lets begin

Here, we have f(x) = xe^x·e^x. Using the product rule, f'(x) = u'v + uv' In the given equation, u = xe^x and v = e^x. Let’s differentiate each term, u' = d/dx (xe^x) = (x + 1)e^x v' = d/dx (e^x) = e^x Substituting into the given equation, f'(x) = [(x + 1)e^x]·e^x + xe^x·e^x Let’s simplify terms to get the final answer, f'(x) = (x + 1)e^(2x) + xe^(2x) = (2x + 1)e^(2x) Thus, the derivative of the specified function is (2x + 1)e^(2x).

Explanation

We find the derivative of the given function by dividing the function into two parts. The first step is finding its derivative and then combining them using the product rule to get the final result.

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Problem 2

The city of Greendale is constructing a new bridge. The elevation is represented by the function y = xe^x, where y represents the height of the bridge at a distance x. If x = 2 meters, measure the rate of change of the bridge's elevation.

Okay, lets begin

We have y = xe^x (elevation of the bridge)...(1) Now, we will differentiate the equation (1) Take the derivative of xe^x: dy/dx = (x + 1)e^x Given x = 2 (substitute this into the derivative) dy/dx = (2 + 1)e^2 dy/dx = 3e^2 Hence, the rate of change of the bridge's elevation at x = 2 meters is 3e^2.

Explanation

We find the rate of change of the bridge's elevation at x = 2 meters as 3e^2, which means that at a given point, the height of the bridge would increase at a rate of 3e^2 times the horizontal distance.

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Problem 3

Derive the second derivative of the function y = xe^x.

Okay, lets begin

The first step is to find the first derivative, dy/dx = (x + 1)e^x...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [(x + 1)e^x] Here we use the product rule again, d²y/dx² = [(x + 1)'e^x + (x + 1)d/dx(e^x)] = [1·e^x + (x + 1)e^x] = e^x + (x + 1)e^x = (x + 2)e^x Therefore, the second derivative of the function y = xe^x is (x + 2)e^x.

Explanation

We use the step-by-step process, where we start with the first derivative. Using the product rule, we differentiate (x + 1)e^x. We then substitute the identity and simplify the terms to find the final answer.

Well explained 👍

Problem 4

Prove: d/dx (x²e^x) = (x² + 2x)e^x.

Okay, lets begin

Let’s start using the product rule: Consider y = x²e^x y = (x²)·(e^x) To differentiate, we use the product rule: dy/dx = d/dx(x²)·e^x + x²·d/dx(e^x) Since the derivative of x² is 2x and the derivative of e^x is e^x, dy/dx = 2x·e^x + x²·e^x dy/dx = (2x + x²)e^x Substituting y = x²e^x, d/dx (x²e^x) = (x² + 2x)e^x Hence proved.

Explanation

In this step-by-step process, we used the product rule to differentiate the equation. Then, we replace each component with its derivative. As a final step, we substitute y = x²e^x to derive the equation.

Well explained 👍

Problem 5

Solve: d/dx (xe^x/x)

Okay, lets begin

To differentiate the function, we simplify it first: d/dx (xe^x/x) = d/dx (e^x) Since the x terms cancel out, the derivative of e^x is simply e^x. Therefore, d/dx (xe^x/x) = e^x.

Explanation

In this process, we simplify the given function by canceling out x terms, which allows us to easily find the derivative. The result is straightforward as the derivative of e^x is e^x.

Well explained 👍

FAQs on the Derivative of xe^x

1.Find the derivative of xe^x.

Using the product rule on xe^x gives: d/dx (xe^x) = (x + 1)e^x.

2.Can we use the derivative of xe^x in real life?

Yes, we can use the derivative of xe^x in real life in calculating growth rates, especially in fields such as biology, physics, and economics.

3.Is it possible to take the derivative of xe^x at x = 0?

Yes, at x = 0, the derivative of xe^x is e^0, which equals 1.

4.What rule is used to differentiate xe^x?

We use the product rule to differentiate xe^x, where the derivative is (x + 1)e^x.

5.Are the derivatives of xe^x and x(e^x) the same?

Yes, they are the same since the expression x(e^x) is equivalent to xe^x. The derivative for both is (x + 1)e^x.

Important Glossaries for the Derivative of xe^x

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Exponential Function: A function of the form e^x, where e is Euler's number, a mathematical constant approximately equal to 2.71828. Product Rule: A rule used to find the derivative of the product of two functions. First Derivative: It is the initial result of a function, which gives us the rate of change of a specific function. Chain Rule: A rule used to differentiate compositions of functions.

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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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