Derivative Of (3x + 1) / (2x - 1) Explained Step-by-Step

Jun 27, 2025 by ADMIN 57 views

Find the derivative of (3x + 1) / (2x - 1)

Exploring the Derivative of (3x + 1) / (2x - 1): A Comprehensive Guide

Introduction: Understanding Derivatives

In the realm of calculus, derivatives play a crucial role in understanding the rate at which a function's output changes with respect to its input. The derivative, often denoted as d/dx, provides the instantaneous rate of change, which is essential in various fields, including physics, engineering, economics, and computer science. Mastering the techniques of differentiation is paramount for anyone seeking to delve deeper into these disciplines. In this article, we will explore the process of finding the derivative of the function (3x + 1) / (2x - 1), a classic example that demonstrates the application of the quotient rule. This rule is a fundamental tool in calculus for differentiating functions that are expressed as the ratio of two other functions. By understanding the steps involved in this process, you will gain a solid foundation for tackling more complex differentiation problems.

The derivative of a function, at its core, represents the slope of the tangent line to the function's graph at a specific point. This geometric interpretation provides a visual understanding of how the function is behaving at that particular point. A positive derivative indicates that the function is increasing, a negative derivative signifies a decreasing function, and a derivative of zero implies a stationary point, where the function is neither increasing nor decreasing. These critical points are often the locations of local maxima or minima, which are crucial in optimization problems. For instance, in physics, the derivative of a position function with respect to time gives the velocity, and the derivative of the velocity function gives the acceleration. In economics, derivatives are used to determine marginal cost and marginal revenue, which are essential for making informed business decisions. The power of derivatives extends to numerous real-world applications, making it an indispensable tool for problem-solving and analysis.

The process of finding a derivative involves applying a set of rules and techniques that have been developed over centuries. These rules allow us to systematically differentiate various types of functions, from simple polynomials to complex trigonometric and exponential functions. The quotient rule, which we will use in this article, is one such technique specifically designed for functions that are expressed as a fraction. Other important rules include the power rule, the product rule, and the chain rule, each catering to different functional forms. Understanding these rules and knowing when to apply them is key to successfully differentiating a wide range of functions. Furthermore, the ability to manipulate and simplify algebraic expressions is crucial in the process of differentiation, as it often makes the application of these rules more straightforward. By mastering these fundamental concepts and techniques, you will be well-equipped to tackle the challenges of differential calculus and its applications.

Problem Statement: Finding d/dx((3x + 1) / (2x - 1))

Our primary goal is to determine the derivative of the function (3x + 1) / (2x - 1) with respect to x, which is mathematically represented as d/dx((3x + 1) / (2x - 1)). This expression signifies the instantaneous rate of change of the function as x varies. To achieve this, we will employ the quotient rule, a powerful tool in calculus specifically designed for differentiating functions that are expressed as the ratio of two other functions. The quotient rule provides a systematic approach to handle such derivatives, ensuring accuracy and efficiency. By applying this rule, we can break down the complex derivative into manageable parts and arrive at the solution. Understanding the quotient rule and its application is fundamental for mastering differentiation techniques and solving a wide range of calculus problems. This specific problem serves as an excellent example to illustrate the practical use of the quotient rule and its importance in calculus.

The function (3x + 1) / (2x - 1) is a rational function, which is a function that can be expressed as the ratio of two polynomials. Rational functions are common in many areas of mathematics and physics, and their derivatives often play a critical role in understanding their behavior. For instance, the derivative can help us identify the critical points of the function, such as local maxima and minima, and determine the intervals where the function is increasing or decreasing. These insights are invaluable in various applications, such as optimization problems and curve sketching. Moreover, rational functions often arise in modeling real-world phenomena, making their derivatives essential in understanding and predicting the behavior of these systems. The quotient rule is particularly well-suited for differentiating rational functions, as it directly addresses the challenge of differentiating a ratio of two expressions. By mastering the application of the quotient rule to functions like (3x + 1) / (2x - 1), you will gain a valuable skill that is applicable in many different contexts.

Before diving into the solution, it's important to recognize the structure of the function and understand why the quotient rule is the appropriate tool. The function is clearly a fraction, with the numerator being (3x + 1) and the denominator being (2x - 1). Both the numerator and the denominator are linear functions, which are relatively simple to differentiate individually. However, the derivative of the entire fraction is not simply the ratio of the derivatives of the numerator and denominator. This is where the quotient rule comes into play, providing the correct formula for differentiating such functions. The quotient rule takes into account the interaction between the numerator and the denominator and ensures that the derivative is calculated accurately. By understanding the rationale behind using the quotient rule in this context, you will gain a deeper appreciation for its importance and applicability in calculus.

Applying the Quotient Rule: Step-by-Step Solution

The quotient rule is a fundamental theorem in calculus that provides a method for finding the derivative of a function that is expressed as the ratio of two differentiable functions. It states that if we have a function h(x) = f(x) / g(x), where f(x) and g(x) are differentiable functions, then the derivative of h(x) with respect to x, denoted as h'(x), is given by the formula: h'(x) = [g(x) * f'(x) - f(x) * g'(x)] / [g(x)]^2. This formula is crucial for differentiating rational functions, where the numerator and denominator are both functions of x. The quotient rule ensures that we account for the interplay between the numerator and the denominator when calculating the derivative, which is essential for obtaining the correct result. Understanding and applying the quotient rule is a key skill in differential calculus, enabling us to handle a wide variety of differentiation problems involving fractions.

To apply the quotient rule to our specific problem, d/dx((3x + 1) / (2x - 1)), we first identify the functions f(x) and g(x). In this case, f(x) = 3x + 1 is the numerator, and g(x) = 2x - 1 is the denominator. Next, we need to find the derivatives of f(x) and g(x) with respect to x. The derivative of f(x), denoted as f'(x), is the derivative of 3x + 1, which is simply 3. This is because the derivative of 3x is 3, and the derivative of the constant 1 is 0. Similarly, the derivative of g(x), denoted as g'(x), is the derivative of 2x - 1, which is 2. Again, the derivative of 2x is 2, and the derivative of the constant -1 is 0. Now that we have identified f(x), g(x), f'(x), and g'(x), we can plug these into the quotient rule formula to find the derivative of the given function. This step-by-step approach ensures that we apply the quotient rule correctly and systematically, minimizing the chances of making errors.

Now, let's substitute f(x) = 3x + 1, g(x) = 2x - 1, f'(x) = 3, and g'(x) = 2 into the quotient rule formula: h'(x) = [g(x) * f'(x) - f(x) * g'(x)] / [g(x)]^2. Plugging in the values, we get h'(x) = [(2x - 1) * 3 - (3x + 1) * 2] / (2x - 1)^2. The next step is to simplify the expression. First, we expand the terms in the numerator: (2x - 1) * 3 = 6x - 3 and (3x + 1) * 2 = 6x + 2. Substituting these back into the expression, we have h'(x) = [6x - 3 - (6x + 2)] / (2x - 1)^2. Next, we distribute the negative sign in the numerator: h'(x) = [6x - 3 - 6x - 2] / (2x - 1)^2. We can now combine like terms in the numerator: 6x - 6x cancels out, and -3 - 2 = -5. Therefore, the derivative simplifies to h'(x) = -5 / (2x - 1)^2. This is the final derivative of the given function, obtained by systematically applying the quotient rule and simplifying the resulting expression.

Solution: d/dx((3x + 1) / (2x - 1)) = -5 / (2x - 1)^2

After meticulously applying the quotient rule and simplifying the resulting expression, we have arrived at the solution: d/dx((3x + 1) / (2x - 1)) = -5 / (2x - 1)^2. This result represents the instantaneous rate of change of the function (3x + 1) / (2x - 1) with respect to x. The negative sign in the numerator indicates that the function is decreasing over its domain, while the denominator (2x - 1)^2 shows how the rate of change varies as x changes. The denominator also implies that the function is undefined at x = 1/2, as this would result in division by zero. This final derivative provides valuable information about the behavior of the original function, allowing us to analyze its properties and apply it in various contexts. Understanding how to arrive at this solution through the quotient rule is a crucial skill in calculus, demonstrating the power and utility of this fundamental theorem.

This derivative can be further analyzed to understand the behavior of the original function. For instance, we can observe that the derivative is always negative (except at x = 1/2 where it is undefined) because the numerator is -5, and the denominator (2x - 1)^2 is always positive (or zero) for any real number x. This indicates that the function (3x + 1) / (2x - 1) is decreasing over its entire domain, except at the point x = 1/2 where it has a vertical asymptote. The rate of decrease is more pronounced when x is close to 1/2, as the denominator becomes smaller, and the derivative becomes larger in magnitude. Conversely, as x moves away from 1/2, the rate of decrease becomes less pronounced. This analysis provides a comprehensive understanding of how the function changes and behaves across its domain, illustrating the power of derivatives in function analysis. By examining the derivative, we can gain insights into the original function that would not be immediately apparent from the function itself.

In conclusion, finding the derivative of (3x + 1) / (2x - 1) using the quotient rule not only provides the solution -5 / (2x - 1)^2 but also enhances our understanding of the function's behavior. The quotient rule is a vital tool in calculus, enabling us to differentiate functions expressed as ratios. By mastering this technique and understanding the implications of the derivative, we can tackle more complex problems and gain deeper insights into mathematical and real-world phenomena. This specific example serves as a clear illustration of the quotient rule's application and the valuable information that derivatives can provide about a function's properties. The ability to differentiate functions effectively is a cornerstone of calculus and is essential for anyone seeking to delve deeper into mathematics, physics, engineering, and other related fields.

Practice Problems: Further Exploration

To solidify your understanding of the quotient rule and its application, it is essential to practice with a variety of problems. Working through different examples will help you become more comfortable with the rule and its nuances. Practice problems allow you to apply the theoretical knowledge you have gained and develop practical skills in differentiation. By tackling different functions and scenarios, you will also learn to recognize when the quotient rule is the appropriate technique to use. This hands-on experience is crucial for mastering calculus and building confidence in your problem-solving abilities. Furthermore, practice problems can expose you to different types of functions and algebraic manipulations, enhancing your overall mathematical proficiency.

Here are a few additional practice problems that you can attempt to further enhance your understanding of the quotient rule:

Find the derivative of (x^2 + 1) / (x - 1)
Determine d/dx of (sin(x)) / (cos(x))
Calculate the derivative of (e^x) / (x + 1)

These problems offer a range of complexity and involve different types of functions, including polynomials, trigonometric functions, and exponential functions. Attempting these problems will not only reinforce your understanding of the quotient rule but also help you develop a broader skillset in differentiation. For each problem, remember to identify the functions f(x) and g(x), find their derivatives f'(x) and g'(x), and then apply the quotient rule formula. Be sure to simplify your final answer as much as possible. Working through these practice problems will significantly contribute to your mastery of the quotient rule and your overall success in calculus.

By consistently practicing and applying the quotient rule, you will develop a strong foundation in differentiation techniques. This skill is not only essential for calculus but also for many other areas of mathematics and science. The ability to confidently differentiate functions will open doors to more advanced topics and applications, allowing you to tackle complex problems and gain deeper insights into the world around you. So, take the time to work through these practice problems and continue to explore the fascinating world of calculus!