Finding The Derivative Of F(x) = 5/x³ A Step-by-Step Guide
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In this comprehensive guide, we will delve into the process of determining the derivative of the function f(x) = 5/x³. This is a fundamental concept in calculus, and understanding how to find derivatives is crucial for various applications in mathematics, physics, engineering, and other fields. We will break down the problem step-by-step, explaining the underlying principles and techniques involved. Whether you're a student learning calculus for the first time or someone looking to refresh your knowledge, this guide will provide you with a clear and thorough explanation.
Before we dive into the specific problem, let's briefly review what a derivative is. In simple terms, the derivative of a function f(x) represents the instantaneous rate of change of the function with respect to its input variable x. Geometrically, it represents the slope of the tangent line to the graph of the function at a given point. The derivative is denoted by f'(x) or df/dx. The derivative is a fundamental concept in calculus, serving as a cornerstone for understanding rates of change and optimization problems. Its applications span diverse fields, including physics, engineering, economics, and computer science. In physics, derivatives are used to calculate velocity and acceleration. In engineering, they aid in designing structures and systems. In economics, derivatives help model economic growth and predict market behavior. In computer science, they are essential for machine learning algorithms. The concept of a derivative allows us to analyze how a function changes at any given point, providing valuable insights into its behavior and characteristics. Understanding derivatives is crucial for anyone seeking a deeper understanding of mathematical and scientific principles. Mastering the techniques for calculating derivatives opens doors to solving a wide range of real-world problems and making informed decisions based on mathematical models. For example, in finance, derivatives can be used to assess risk and optimize investment strategies. In biology, they can model population growth and predict the spread of diseases. The versatility of derivatives makes them an indispensable tool in many disciplines. We will see how derivatives enable us to analyze and interpret complex phenomena. By examining the derivative, we can pinpoint critical points of a function, such as maxima and minima, which are vital in optimization problems. This is particularly useful in business and engineering, where maximizing profits or minimizing costs is a primary goal. Furthermore, derivatives help us to understand the concavity of a function, which indicates whether the function is increasing or decreasing at an accelerating or decelerating rate. This information is crucial for sketching accurate graphs of functions and visualizing their behavior. The power rule, the chain rule, and the product rule are just a few of the many techniques available for finding derivatives. Each rule applies to a specific type of function or combination of functions. The power rule is fundamental for differentiating polynomial functions, while the chain rule is essential for composite functions. The product rule handles functions that are products of two or more simpler functions. By mastering these rules and understanding when to apply them, one can confidently tackle a wide array of differentiation problems. The derivative not only provides information about the slope of a function but also about its behavior over intervals. For example, if the derivative is positive over an interval, the function is increasing over that interval. If the derivative is negative, the function is decreasing. These insights are invaluable for understanding the overall trend of a function and making predictions about its future values. Derivatives also play a central role in optimization problems, where the goal is to find the maximum or minimum value of a function subject to certain constraints. By setting the derivative equal to zero and solving for the critical points, we can identify potential maxima and minima. Further analysis, such as the second derivative test, can help us determine whether these points are indeed maxima, minima, or saddle points. These optimization techniques are widely used in various fields, from engineering design to financial modeling. Derivatives are an essential tool for understanding and modeling the dynamic behavior of systems. They allow us to analyze how quantities change over time and make predictions about future states. Whether it's modeling the motion of a projectile, the growth of a population, or the flow of current in an electrical circuit, derivatives provide the mathematical framework for capturing these dynamic processes. In essence, the derivative is more than just a mathematical concept; it is a powerful tool for understanding the world around us. Its applications are vast and varied, making it an indispensable part of the toolkit of scientists, engineers, and anyone who seeks to analyze and solve complex problems. The ability to calculate and interpret derivatives is a valuable skill that can lead to deeper insights and more informed decision-making.
Our function is f(x) = 5/x³. To make it easier to differentiate, we can rewrite this function using negative exponents. Recall that 1/xⁿ = x⁻ⁿ. Therefore, we can rewrite f(x) as:
f(x) = 5x⁻³
This form is much more convenient for applying the power rule of differentiation, which we will discuss next. Rewriting the function in this way not only simplifies the differentiation process but also provides a clearer view of the function's structure. By expressing the function with a negative exponent, we can directly apply the power rule without needing to deal with fractions. This transformation is a common technique used in calculus to make differentiation easier and more efficient. The use of negative exponents allows us to treat the function as a simple power function, which is readily handled by the power rule. This technique is especially useful when dealing with functions that involve rational expressions or reciprocals. By transforming the function into a power function form, we can avoid complex algebraic manipulations and focus on the core differentiation process. This step is crucial as it sets the stage for applying the power rule directly and efficiently. The transformed function f(x) = 5x⁻³ is now in a form that is easily recognizable and compatible with the power rule, making the subsequent differentiation steps straightforward. This rewriting not only simplifies the mathematical process but also reduces the chance of making errors. By avoiding complex fractions and rational expressions, we can focus on the key steps involved in differentiation. This technique is a valuable tool in the calculus toolkit, enabling us to handle a wide range of functions with greater ease and accuracy. In addition to simplifying differentiation, rewriting the function with negative exponents also helps in understanding the function's behavior. The negative exponent indicates that the function has a reciprocal relationship with x, meaning that as x increases, the function decreases, and vice versa. This insight can be valuable in analyzing the function's graph and predicting its behavior. Furthermore, the rewritten form makes it easier to identify the function's domain and any potential discontinuities. For instance, we can see that the function is not defined at x = 0 due to the presence of the negative exponent. This type of analysis is crucial for a comprehensive understanding of the function and its properties. The ability to rewrite functions in different forms is a fundamental skill in calculus. It allows us to manipulate expressions to make them more amenable to mathematical operations. This skill is not only useful for differentiation but also for integration, simplification, and other calculus techniques. Mastering these transformations can significantly enhance our ability to solve complex mathematical problems and gain deeper insights into the functions we are working with. The transformation from a rational expression to a power function is a common and powerful technique in calculus. It allows us to apply the well-established rules of differentiation to a broader range of functions. This technique is not limited to simple rational functions like 5/x³; it can be applied to more complex expressions as well. By becoming proficient in this transformation, we can significantly expand our capabilities in calculus and tackle more challenging problems. The rewritten form of the function is now perfectly poised for the application of the power rule, which we will explore in the next section. This step-by-step approach ensures that we understand each aspect of the differentiation process, making the solution clear and accessible. The process of rewriting the function also highlights the importance of algebraic manipulation in calculus. Being able to manipulate expressions to fit the appropriate form for differentiation is a crucial skill. It allows us to apply standard differentiation rules without getting bogged down in complex algebraic manipulations. This skill is honed through practice and exposure to a variety of functions, and it forms the backbone of many calculus techniques.
The power rule of differentiation states that if f(x) = axⁿ, where a is a constant and n is any real number, then the derivative f'(x) is given by:
f'(x) = naxⁿ⁻¹
In our case, f(x) = 5x⁻³. Here, a = 5 and n = -3. Applying the power rule, we get:
f'(x) = (-3)(5)x⁻³⁻¹ f'(x) = -15x⁻⁴
The power rule is a cornerstone of differential calculus, providing a straightforward method for finding the derivatives of power functions. Its simplicity and broad applicability make it an essential tool for anyone studying calculus. The rule states that if you have a function of the form f(x) = axⁿ, where a is a constant and n is a real number, the derivative f'(x) is obtained by multiplying the coefficient a by the exponent n and then reducing the exponent by 1. This rule is fundamental because it covers a wide range of functions, including polynomials and functions with fractional or negative exponents. The power rule's elegance lies in its ability to transform a function into its derivative in a systematic and predictable way. It is a direct application of the definition of the derivative, which involves finding the limit of the difference quotient. However, the power rule provides a shortcut that avoids the need to compute limits directly, making differentiation much more efficient. The power rule is not just a theoretical concept; it has numerous practical applications. In physics, for example, it is used to find the velocity and acceleration of an object given its position as a function of time. In economics, it can be used to model the rate of change of economic variables. In engineering, it helps in optimizing designs and processes. Understanding the power rule also lays the foundation for learning more advanced differentiation techniques, such as the product rule, quotient rule, and chain rule. These rules build upon the power rule and allow us to differentiate more complex functions. The power rule's simplicity can sometimes be deceptive. It's important to remember that the rule applies directly only to functions in the form axⁿ. If a function is not in this form, it may need to be rewritten or simplified before the power rule can be applied. This often involves algebraic manipulations such as expanding expressions, combining terms, or using trigonometric identities. One of the most common applications of the power rule is in differentiating polynomials. A polynomial is a function of the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where the coefficients aᵢ are constants and the exponents are non-negative integers. To differentiate a polynomial, we simply apply the power rule to each term individually and then add the results. The power rule is also indispensable when dealing with functions that involve radicals. For example, the square root function f(x) = √x can be rewritten as f(x) = x¹/², which is in the form axⁿ with a = 1 and n = 1/2. Applying the power rule, we find that the derivative is f'(x) = (1/2)x⁻¹/², which can be rewritten as f'(x) = 1/(2√x). This demonstrates the power rule's versatility in handling fractional exponents. In our specific problem, f(x) = 5x⁻³, we identified a = 5 and n = -3. Applying the power rule, we multiplied the coefficient 5 by the exponent -3 to get -15, and then we reduced the exponent by 1 to get -4. This gave us the derivative f'(x) = -15x⁻⁴. This example clearly illustrates the step-by-step application of the power rule. The power rule is a fundamental building block in calculus. Its mastery is essential for anyone seeking to progress in the field. By understanding the power rule and its applications, we can confidently differentiate a wide range of functions and solve various problems involving rates of change and optimization. The power rule not only simplifies differentiation but also enhances our understanding of the behavior of functions. By examining the derivative, we can infer valuable information about the original function, such as its increasing and decreasing intervals, its concavity, and its critical points. This connection between the function and its derivative is a central theme in calculus. The process of applying the power rule reinforces the importance of paying attention to detail and following a systematic approach. Each step in the calculation must be performed carefully to avoid errors. This methodical approach is a hallmark of mathematical problem-solving and is applicable to a wide range of situations. In conclusion, the power rule is a cornerstone of differential calculus. Its mastery is crucial for anyone seeking to understand and apply the concepts of calculus. The rule's simplicity and broad applicability make it an indispensable tool for solving a wide range of problems in mathematics, science, and engineering.
We have found that f'(x) = -15x⁻⁴. To express this derivative with a positive exponent, we can rewrite it as:
f'(x) = -15/x⁴
This is the simplified form of the derivative of the given function. Simplifying the derivative is a crucial step in calculus, as it often makes the result more readable and easier to work with in subsequent calculations. In our case, we obtained f'(x) = -15x⁻⁴ after applying the power rule. While this is a correct representation of the derivative, it is customary to express results with positive exponents whenever possible. Simplifying the derivative not only presents the answer in a more conventional form but also provides a clearer understanding of the function's behavior. A negative exponent indicates a reciprocal relationship, which can sometimes be less intuitive than a positive exponent. By rewriting x⁻⁴ as 1/x⁴, we make the reciprocal relationship explicit, allowing for easier interpretation. For example, we can immediately see that as x increases in magnitude, the term 1/x⁴ decreases, and thus the derivative f'(x) approaches zero. This information can be valuable in sketching the graph of the derivative and understanding the rate of change of the original function. Simplification is not just about aesthetics; it also plays a practical role in further mathematical operations. For instance, if we need to find the second derivative or use the derivative in an optimization problem, a simplified form will make the calculations less cumbersome. A simplified derivative can also facilitate the process of finding critical points, intervals of increase and decrease, and concavity. In many applications of calculus, such as finding the equation of a tangent line or determining the maximum or minimum value of a function, a simplified derivative is essential. It allows for accurate and efficient calculations, reducing the likelihood of errors. The process of simplifying often involves rewriting expressions using algebraic rules and identities. In this case, we used the rule x⁻ⁿ = 1/xⁿ to convert the negative exponent to a positive exponent. Other common simplification techniques include combining like terms, factoring expressions, and using trigonometric identities. Mastering these techniques is crucial for success in calculus and related fields. Simplifying a derivative also highlights the importance of mathematical notation and conventions. While f'(x) = -15x⁻⁴ and f'(x) = -15/x⁴ are mathematically equivalent, the latter is generally preferred due to its clarity and ease of interpretation. Adhering to standard notation conventions makes mathematical communication more effective and reduces the potential for misunderstanding. The simplified form of the derivative also provides a more direct connection to the original function. By looking at f'(x) = -15/x⁴, we can see that the derivative is always negative (except at x = 0, where it is undefined), which indicates that the original function f(x) = 5/x³ is decreasing for all x in its domain. This type of insight is valuable in understanding the behavior of functions and their derivatives. Simplifying a derivative is an essential skill that goes beyond just finding the correct answer. It is about presenting the answer in the most informative and usable form. This skill is honed through practice and attention to detail, and it is a hallmark of a strong calculus student. The act of simplification also reinforces the importance of precision and accuracy in mathematical calculations. Each step in the simplification process must be performed carefully to avoid errors. This meticulous approach is crucial for success in mathematics and other quantitative fields. In conclusion, simplifying the derivative is a critical step in calculus that enhances clarity, facilitates further calculations, and provides valuable insights into the behavior of functions. It is a skill that is well worth mastering, as it will significantly improve one's ability to solve calculus problems and communicate mathematical ideas effectively.
In this guide, we have successfully determined the derivative of the function f(x) = 5/x³. We rewrote the function as f(x) = 5x⁻³, applied the power rule to find the derivative f'(x) = -15x⁻⁴, and simplified the result to f'(x) = -15/x⁴. This step-by-step approach demonstrates the fundamental principles and techniques involved in differentiation. Mastering these concepts is essential for further exploration in calculus and its applications.
