Solving The Equation: X * F'(x) = 2f(x)

Hey guys! Let's dive into solving this math problem together. We've got the equation x * f'(x) = 2f(x), and we're given that f(x) = x³ ln x. Our mission? To find the value of x that satisfies this equation. Sounds like fun, right? Let's break it down step by step. This is a classic calculus problem, so we'll be using derivatives and a bit of algebra to find the solution. Get ready to flex those math muscles!

Understanding the Problem: A Quick Overview

Alright, before we jump into the nitty-gritty, let's make sure we're all on the same page. The equation x * f'(x) = 2f(x) essentially tells us that we need to find a point x where x multiplied by the derivative of f(x) is equal to twice the function f(x) itself. The derivative, denoted by f'(x), represents the rate of change of the function f(x). In our case, f(x) = x³ ln x. This function involves both a power function (x³) and a logarithmic function (ln x). Our goal is to find the specific value(s) of x that make this equation true. We'll need to calculate the derivative of f(x) and then substitute it back into the original equation. The options we're given are: A) 1/e, B) e, C) 1/e², D) 2e, E) 1. We will need to check which of these values satisfies the equation. Remember, the derivative is crucial; it tells us how the function changes at any given point. We will use the product rule and chain rule here. So, let's get started!

This problem is a good test of your understanding of derivatives, algebraic manipulation, and problem-solving skills. The key is to approach it systematically, breaking down the problem into smaller, manageable steps. Don't worry if it seems a bit tricky at first; with practice and a clear understanding of the concepts, you'll be able to solve it with ease. Let's get into the next stage of the problem! We need to figure out how to find the derivative of f(x).

Finding the Derivative f'(x)

Okay, now comes the fun part: finding the derivative of f(x). Since f(x) = x³ ln x, we'll need to use the product rule. The product rule states that if we have a function h(x) = u(x) * v(x), then its derivative h'(x) = u'(x) * v(x) + u(x) * v'(x). In our case, let's consider u(x) = x³ and v(x) = ln x. So, we need to find the derivatives of u(x) and v(x) separately.

First, let's find u'(x), the derivative of x³. Using the power rule (which says that the derivative of xⁿ is n x⁽ⁿ⁻¹⁾), we get u'(x) = 3x². Next, let's find v'(x), the derivative of ln x. The derivative of ln x is 1/x, so v'(x) = 1/x. Now, applying the product rule: f'(x) = u'(x) * v(x) + u(x) * v'(x) = (3x²) * (ln x) + (x³) * (1/x). Simplifying this, we get f'(x) = 3x² ln x + x². We've successfully found the derivative of f(x). Now, we need to plug this into our original equation and solve for x. Don't worry, we're almost there, we're doing great. Just a little bit more work and we'll have our answer!

Remember, finding derivatives is a fundamental skill in calculus, and mastering the product rule is key. Pay close attention to each step to make sure you understand where everything comes from. Always double-check your work. Making small mistakes can lead to the wrong answers. Now that we have the derivatives, we can start by substituting them into the equation. The next step will get us closer to the solution.

Substituting and Solving the Equation

Alright, time to put it all together! We have f'(x) = 3x² ln x + x² and the original equation x * f'(x) = 2f(x), and we know that f(x) = x³ ln x. Now, let's substitute the derivative and the original function into the equation: x * (3x² ln x + x²) = 2 * (x³ ln x). Let's simplify this a bit. First, distribute the x on the left side: 3x³ ln x + x³ = 2x³ ln x. Now, let's gather all the terms on one side of the equation. Subtract 2x³ ln x from both sides: x³ ln x + x³ = 0. We can factor out an x³: x³ (ln x + 1) = 0. This gives us two potential solutions: x³ = 0 or (ln x + 1) = 0.

For x³ = 0, the solution is x = 0. However, ln x is not defined for x = 0. Therefore, x = 0 is not a valid solution in this context. For (ln x + 1) = 0, subtract 1 from both sides: ln x = -1. To solve for x, we can exponentiate both sides using the base e: e^(ln x) = e^(-1). This simplifies to x = 1/e. So, our solution is x = 1/e. Now we know the value of x, we have found the solution to the equation! We're done with the hard part. Let's check our answer against the options provided to us. Let's move on to the next step to choose the right answer.

Remember that logarithms are a fundamental concept in mathematics. Understanding their properties and how they interact with other functions is crucial for solving this type of problem. The substitution step may look complicated at first. However, with practice, you'll quickly get the hang of it. Be patient, and take your time to grasp each step. Don't rush the process, and always double-check your work. Now, let's check our final answer.

Checking the Solution Against the Options

Okay, we've done the hard work, and we've found our solution. Now, let's compare our answer with the given options: A) 1/e, B) e, C) 1/e², D) 2e, E) 1. We calculated that x = 1/e. This matches option A. Therefore, the correct answer is A) 1/e. Congratulations, guys! We solved it! We found the value of x that satisfies the original equation, which is 1/e. We correctly found the solution by calculating the derivative of f(x), substituting it back into the equation, and solving for x.

We carefully considered all the steps, from understanding the problem and finding the derivative to substituting and simplifying the equation. It's a great feeling when everything comes together and you get the correct answer. This problem is a great example of how calculus concepts can be used to solve real-world problems. Keep practicing and you'll get better. Now you know how to solve these types of equations. Good luck, and thanks for joining me!

Conclusion: Final Thoughts

So there you have it! We've successfully navigated the equation x * f'(x) = 2f(x), where f(x) = x³ ln x, and found that x = 1/e. This problem highlights the importance of understanding derivatives, the product rule, and algebraic manipulation. By breaking down the problem into smaller, manageable steps, we were able to find the solution. Remember, practice is key when it comes to mastering calculus. Keep working on problems like this, and you'll become more and more comfortable with these concepts. The journey may seem tough, but the rewards are worth it. Hopefully, this detailed guide has been helpful. If you have any questions, feel free to ask. Happy calculating, and keep exploring the fascinating world of mathematics!

In solving this problem, we not only applied calculus but also strengthened our problem-solving skills. These skills are valuable in various aspects of life, not just in mathematics. Remember to always break down complex problems into smaller parts and approach them systematically. This will help you to understand the concepts thoroughly. We found that using product rule is helpful to compute derivative easily. Overall, the solution to the equation x * f'(x) = 2f(x), where f(x) = x³ ln x, is x = 1/e. We hope that this tutorial was helpful and easy to understand. Keep practicing.