101 LearnersLast updated on September 1, 2025
Derivative of x^e
We use the derivative of x^e, which is e*x^(e-1), as a measuring tool for 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 talk about the derivative of x^e in detail.
What is the Derivative of x^e?
We now understand the derivative of xe.
It is commonly represented as d/dx (xe) or (xe)', and its value is e*x(e-1).
The function xe has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Exponential Function: xe is a power function where the base is x and the exponent is e.
Power Rule: Rule for differentiating functions of the form xn, which applies to xe.
Constant e: The number e is a mathematical constant approximately equal to 2.71828.
Derivative of x^e Formula
The derivative of x^e can be denoted as d/dx (xe) or (xe)'. The formula we use to differentiate xe is: d/dx (xe) = e*x(e-1) The formula applies to all x > 0, given e is a constant.
Proofs of the Derivative of x^e
We can derive the derivative of xe using proofs.
To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as:
Using the Power Rule Using Logarithmic Differentiation We will now demonstrate that the differentiation of xe results in e*x(e-1) using the above-mentioned methods:
Using the Power Rule The derivative of xe can be proved using the Power Rule, which states that d/dx (xn) = n*x(n-1).
Given that our function is f(x) = xe, apply the Power Rule: f'(x) = e*x(e-1)
Using Logarithmic Differentiation To prove the differentiation of x^e using logarithmic differentiation, we take the natural log of both sides.
Let y = xe Take the natural log of both sides: ln(y) = ln(xe) ln(y) = e*ln(x)
Differentiate both sides with respect to x: (1/y) * dy/dx = e/x dy/dx = e*y/x
Since y = xe, substitute back: dy/dx = e*xe/x dy/dx = e*x(e-1)
Hence, proved.
Higher-Order Derivatives of x^e
When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a bit complex.
To understand them better, consider a scenario where acceleration (second derivative) and its rate of change (third derivative) are analyzed.
Higher-order derivatives help us understand functions like xe in greater depth.
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, we generally use f(n)(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 = 0, the derivative is undefined because xe is only defined for positive x.
When x = 1, the derivative of xe = e*1(e-1), which simplifies to e.
Common Mistakes and How to Avoid Them in Derivatives of x^e
Students frequently make mistakes when differentiating xe. These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:
Mistake 1
Not simplifying the equation
Students may forget to simplify the equation, which can lead to incomplete or incorrect results. They often skip steps and directly arrive at the result, especially when solving using logarithmic differentiation. Ensure that each step is written in order. Students might think it is tedious, but it is important to avoid errors in the process.
Mistake 2
Forgetting the Domain of xe
They might not remember that xe is undefined for non-positive x when e is not an integer. Keep in mind that you should consider the domain of the function that you differentiate. It will help you understand that the function is not continuous at such certain points.
Mistake 3
Incorrect use of Power Rule
While differentiating functions such as xe, students misapply the power rule.
For example: Incorrect differentiation: d/dx (xe) = (e-1)*x(e-2). The correct rule is d/dx (xn) = n*x(n-1). To avoid this mistake, write the power rule correctly. Always check for errors in the calculation and ensure it is properly simplified.
Mistake 4
Not writing Constants and Coefficients
There is a common mistake that students at times forget to multiply the constants placed before xe.
For example, they incorrectly write d/dx (5xe) = e*x(e-1). Students should check the constants in the terms and ensure they are multiplied properly.
For example, the correct equation is d/dx (5xe) = 5e*x(e-1).
Mistake 5
Not Applying Logarithmic Differentiation
Students often forget to use logarithmic differentiation when necessary. This happens when the derivative of complex power functions is not considered. To fix this error, students should apply logarithmic differentiation to simplify the process.
For example, for functions like x(e*x), logarithmic differentiation can be more efficient.
Examples Using the Derivative of x^e
Problem 1
Calculate the derivative of (x^e * x^2).
Here, we have f(x) = xe * x2.
Using the product rule, f'(x) = u′v + uv′ In the given equation, u = xe and v = x2.
Let’s differentiate each term, u′ = d/dx (xe) = e*x(e-1)
v′ = d/dx (x2) = 2x substituting into the given equation, f'(x) = (e*x(e-1)) * (x2) + (xe) * (2x)
Let’s simplify terms to get the final answer, f'(x) = e*x(e+1) + 2x(e+1)
Thus, the derivative of the specified function is e*x(e+1) + 2x(e+1).
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.
Problem 2
A car accelerates such that its velocity is represented by the function v = x^e, where x represents time in seconds. If x = 2 seconds, find the acceleration of the car.
We have v = xe (velocity of the car)...(1)
Now, we will differentiate the equation (1)
Take the derivative of xe: dv/dx = e*x(e-1)
Given x = 2 (substitute this into the derivative) dv/dx = e*2(e-1)
Therefore, the acceleration of the car at x = 2 seconds is e*2(e-1).
Explanation
We find the acceleration of the car at x = 2 seconds, which means that at this point in time, the rate of change of velocity is e times 2 raised to the power of (e-1).
Problem 3
Derive the second derivative of the function y = x^e.
The first step is to find the first derivative, dy/dx = e*x(e-1)...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [e*x(e-1)]
d²y/dx² = e*(e-1)*x(e-2)
Therefore, the second derivative of the function y = xe is e*(e-1)*x(e-2).
Explanation
We use the step-by-step process, where we start with the first derivative. Using the power rule, we differentiate e*x(e-1). We then simplify the terms to find the final answer.
Problem 4
Prove: d/dx ((x^e)^2) = 2e*x^(2e-1).
Let’s start using the chain rule:
Consider y = (xe)2
To differentiate, we use the chain rule:
dy/dx = 2*(xe)*d/dx[xe]
Since the derivative of xe is e*x(e-1), dy/dx = 2*(xe)*(e*x(e-1))
Substituting y = (xe)2, d/dx ((xe)2) = 2e*x(2e-1)
Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace xe with its derivative. As a final step, we perform the multiplication to derive the equation.
Problem 5
Solve: d/dx (x^e/x)
To differentiate the function, we use the quotient rule:
d/dx (xe/x) = (d/dx (xe) * x - xe * d/dx(x))/x²
We will substitute d/dx (xe) = e*x(e-1) and d/dx (x) = 1 = (e*x(e-1) * x - xe * 1) / x² = (e*xe - xe) / x² = (e-1)*x(e-1)/x
Therefore, d/dx (xe/x) = (e-1)*x(e-1)/x
Explanation
In this process, we differentiate the given function using the quotient rule. As a final step, we simplify the equation to obtain the final result.
FAQs on the Derivative of x^e
1.Find the derivative of x^e.
2.Can we use the derivative of x^e in real life?
3.Is it possible to take the derivative of x^e at the point where x = 0?
4.What rule is used to differentiate x^e/x?
5.Are the derivatives of x^e and x^e different for different values of e?
6.Can we find the derivative of the x^e formula?
Important Glossaries for the Derivative of x^e
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Exponential Function: A function where a constant base is raised to a variable exponent, such as xe.
- Power Rule: A basic rule of differentiation used to find the derivative of power functions like xn.
- Constant e: A mathematical constant approximately equal to 2.71828, often used in exponential growth calculations.
- Logarithmic Differentiation: A technique used to differentiate expressions by taking the natural log of both sides first.
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.
