Difference Quotient Simplified Find For F(x) = 1/(3x)

In calculus, the difference quotient is a fundamental concept that forms the basis for understanding derivatives. It represents the average rate of change of a function over a small interval. The difference quotient is defined as (f(x+h) - f(x)) / h, where f(x) is the function, x is the point at which the rate of change is being calculated, and h is a small change in x. Simplifying the difference quotient is a crucial step in finding the derivative of a function, which measures the instantaneous rate of change. In this article, we will walk through the process of finding and simplifying the difference quotient for the function f(x) = 1/(3x). This process involves algebraic manipulation and simplification, which are essential skills in calculus. We will explore each step in detail, ensuring a clear understanding of the underlying concepts. This article aims to provide a comprehensive guide for students and anyone interested in calculus, making the process of finding and simplifying difference quotients accessible and straightforward. By the end of this guide, you will be equipped with the knowledge and skills to tackle similar problems with confidence. Understanding the difference quotient is not only vital for calculus but also for various applications in physics, engineering, and economics, where rates of change are crucial. So, let's dive in and unravel the intricacies of the difference quotient for the function f(x) = 1/(3x).

Understanding the Difference Quotient

Before we dive into the specific example, let's solidify our understanding of the difference quotient. The difference quotient, mathematically expressed as [f(x+h) - f(x)] / h, is a measure of the average rate of change of a function f(x) over an interval of length h. This concept is foundational in calculus, as it provides the stepping stone to understanding the derivative, which represents the instantaneous rate of change. To truly grasp the significance of the difference quotient, it's helpful to visualize it graphically. Imagine a curve representing the function f(x) on a coordinate plane. If we pick two points on this curve, (x, f(x)) and (x+h, f(x+h)), the difference quotient represents the slope of the secant line that passes through these two points. As h approaches zero, these two points get closer and closer together, and the secant line approaches the tangent line at the point (x, f(x)). The slope of this tangent line is the derivative of the function at that point. The difference quotient formula involves three key components: f(x+h), f(x), and h. Each of these plays a crucial role in calculating the average rate of change. f(x+h) represents the value of the function at the point x+h, while f(x) represents the value of the function at the point x. The difference between these two values, f(x+h) - f(x), gives us the change in the function's value over the interval of length h. Dividing this change by h gives us the average rate of change per unit interval. In essence, the difference quotient quantifies how much the function's value changes relative to the change in the input variable. This understanding is crucial for various applications, including physics, engineering, and economics, where rates of change are fundamental concepts. In the following sections, we'll apply this understanding to a specific function and see how the difference quotient is calculated and simplified in practice.

Applying the Difference Quotient to f(x) = 1/(3x)

Now, let's apply the concept of the difference quotient to the given function, f(x) = 1/(3x). This process involves several steps, including substituting (x+h) into the function, setting up the difference quotient formula, and simplifying the resulting expression. The first step is to find f(x+h). This means replacing every instance of x in the function's expression with (x+h). So, for f(x) = 1/(3x), we have f(x+h) = 1/(3(x+h)). This substitution is crucial because it allows us to evaluate the function at a slightly different point, which is necessary for calculating the rate of change. Next, we set up the difference quotient formula using f(x+h) and f(x). The formula is [f(x+h) - f(x)] / h. Substituting the expressions we have, we get [1/(3(x+h)) - 1/(3x)] / h. This expression represents the average rate of change of the function f(x) = 1/(3x) over the interval from x to x+h. Now comes the crucial step of simplifying this expression. Simplifying the difference quotient often involves algebraic manipulation, such as finding common denominators, combining fractions, and canceling terms. In this case, we need to combine the two fractions in the numerator. To do this, we find a common denominator, which is 3x(x+h). Rewriting the fractions with this common denominator, we get [x/(3x(x+h)) - (x+h)/(3x(x+h))] / h. Combining the fractions in the numerator, we have [(x - (x+h))/(3x(x+h))] / h, which simplifies to [-h/(3x(x+h))] / h. The final step in simplifying is to divide by h. This is the same as multiplying by 1/h. So, we have [-h/(3x(x+h))] * (1/h). The h in the numerator and denominator cancels out, leaving us with -1/(3x(x+h)). This simplified expression is the difference quotient for the function f(x) = 1/(3x). It represents the average rate of change of the function over the interval of length h. In the next section, we will delve deeper into the implications of this result and its connection to the derivative of the function.

Step-by-Step Solution

Let's break down the solution into a step-by-step process for clarity. This meticulous approach will not only help in understanding the solution for this specific function but also provide a template for tackling similar problems in the future. Each step is a building block, and understanding them individually makes the whole process less daunting. First, we identify the given function: f(x) = 1/(3x). This is our starting point. We need to find the difference quotient for this function, which means we need to calculate [f(x+h) - f(x)] / h. The next step is to find f(x+h). To do this, we substitute (x+h) for x in the function: f(x+h) = 1/(3(x+h)). This substitution is crucial, as it allows us to compare the function's values at two slightly different points. Now that we have f(x+h), we can set up the difference quotient formula. Substituting f(x+h) and f(x) into the formula, we get [1/(3(x+h)) - 1/(3x)] / h. This expression represents the average rate of change of the function over the interval from x to x+h. The next step is where the algebraic manipulation comes into play. We need to simplify the expression inside the brackets. This involves finding a common denominator for the two fractions. The common denominator is 3x(x+h). Rewriting the fractions with this common denominator, we get [x/(3x(x+h)) - (x+h)/(3x(x+h))] / h. Now we can combine the fractions in the numerator: [(x - (x+h))/(3x(x+h))] / h. Simplifying the numerator, we get [-h/(3x(x+h))] / h. This step is crucial for reducing the complexity of the expression. The final step is to divide by h, which is the same as multiplying by 1/h. So, we have [-h/(3x(x+h))] * (1/h). The h in the numerator and denominator cancels out, leaving us with -1/(3x(x+h)). This is the simplified difference quotient for the function f(x) = 1/(3x). By following these steps meticulously, we have successfully found and simplified the difference quotient. This step-by-step approach is a valuable tool for solving similar problems and building a strong foundation in calculus.

Detailed Explanation of Simplification Steps

The simplification steps are the heart of finding the difference quotient. These steps involve algebraic manipulations that transform a complex expression into a more manageable form. A thorough understanding of these steps is crucial for success in calculus. Let's delve into a detailed explanation of each simplification step in the context of our function, f(x) = 1/(3x). Our starting point after substituting f(x+h) and f(x) into the difference quotient formula is [1/(3(x+h)) - 1/(3x)] / h. The first simplification step is to combine the two fractions in the numerator. To do this, we need to find a common denominator. The common denominator for 3(x+h) and 3x is 3x(x+h). This is the smallest expression that both denominators can divide into evenly. Next, we rewrite each fraction with the common denominator. To rewrite 1/(3(x+h)) with the denominator 3x(x+h), we multiply both the numerator and the denominator by x, resulting in x/(3x(x+h)). Similarly, to rewrite 1/(3x) with the denominator 3x(x+h), we multiply both the numerator and the denominator by (x+h), resulting in (x+h)/(3x(x+h)). Now we can substitute these rewritten fractions back into the expression: [x/(3x(x+h)) - (x+h)/(3x(x+h))] / h. With a common denominator, we can combine the fractions in the numerator: [(x - (x+h))/(3x(x+h))] / h. This step simplifies the expression by reducing two fractions into one. The next simplification step involves simplifying the numerator. We distribute the negative sign in the expression x - (x+h), which gives us x - x - h. The x terms cancel out, leaving us with -h. So the expression becomes [-h/(3x(x+h))] / h. This simplification step is crucial for revealing opportunities for further simplification. The final simplification step is to divide by h. Dividing by h is the same as multiplying by 1/h. So, we have [-h/(3x(x+h))] * (1/h). The h in the numerator and the h in the denominator cancel out, leaving us with -1/(3x(x+h)). This is the simplified difference quotient for the function f(x) = 1/(3x). Each of these simplification steps is a fundamental algebraic manipulation. Mastering these steps is essential for success in calculus and other areas of mathematics.

The Significance of the Result

The result we obtained, -1/(3x(x+h)), is the simplified difference quotient for the function f(x) = 1/(3x). But what does this expression actually tell us? What is its significance in the broader context of calculus and mathematics? The difference quotient, as we've discussed, represents the average rate of change of a function over a small interval. In this case, -1/(3x(x+h)) tells us how the function f(x) = 1/(3x) changes, on average, as x changes by a small amount h. The negative sign indicates that the function is decreasing; as x increases, f(x) decreases. The expression also shows that the rate of change depends on the value of x. The term 3x(x+h) in the denominator means that the rate of change is inversely proportional to the square of x (approximately, when h is very small). This tells us that the function changes more rapidly for smaller values of x and less rapidly for larger values of x. The most crucial aspect of the difference quotient is its connection to the derivative. The derivative of a function, often denoted as f'(x), represents the instantaneous rate of change of the function at a particular point. It's the slope of the tangent line to the function's graph at that point. The derivative is defined as the limit of the difference quotient as h approaches zero. In other words, f'(x) = lim (h->0) [f(x+h) - f(x)] / h. To find the derivative of f(x) = 1/(3x), we would take the limit of our simplified difference quotient, -1/(3x(x+h)), as h approaches zero. As h approaches zero, the term (x+h) approaches x, so the expression -1/(3x(x+h)) approaches -1/(3x^2). Therefore, the derivative of f(x) = 1/(3x) is f'(x) = -1/(3x^2). This derivative gives us the instantaneous rate of change of the function at any point x. The difference quotient, therefore, is a vital stepping stone to understanding the derivative. It provides the foundation for calculating instantaneous rates of change, which are fundamental in many areas of mathematics, science, and engineering. Understanding the significance of the difference quotient not only helps in calculus but also provides a deeper understanding of how functions change and how rates of change are calculated.

Conclusion

In conclusion, we have successfully found and simplified the difference quotient for the function f(x) = 1/(3x). We started by understanding the fundamental concept of the difference quotient as a measure of the average rate of change of a function over an interval. We then applied this concept to the given function, carefully substituting (x+h) into the function and setting up the difference quotient formula. The core of the process involved simplifying the resulting expression through algebraic manipulations, including finding common denominators, combining fractions, and canceling terms. We meticulously walked through each step of the simplification process, highlighting the importance of each algebraic manipulation. The final simplified difference quotient, -1/(3x(x+h)), provides valuable information about the function's behavior. It tells us how the function changes on average as x changes by a small amount h. The negative sign indicates a decreasing function, and the denominator shows that the rate of change depends on the value of x. Most importantly, we discussed the significance of the difference quotient in relation to the derivative. The derivative, which represents the instantaneous rate of change, is defined as the limit of the difference quotient as h approaches zero. By taking this limit, we can find the derivative of the function, which in this case is -1/(3x^2). This connection between the difference quotient and the derivative underscores the fundamental role of the difference quotient in calculus. Understanding the difference quotient is not just about manipulating algebraic expressions; it's about grasping the core concepts of rates of change and derivatives. These concepts are essential for understanding many phenomena in the world around us, from the motion of objects to the growth of populations. This exploration of the difference quotient for f(x) = 1/(3x) serves as a building block for more advanced topics in calculus and provides a solid foundation for further study. The skills and understanding gained here will be invaluable in tackling more complex problems and applications in the future.