Finding The Derivative Of Ln(8 - 5x + 5x³)

In the realm of calculus, finding the derivative of a function is a fundamental operation that reveals crucial information about the function's behavior. This article delves into the process of determining the derivative of the function g(x) = ln(8 - 5x + 5x³), providing a step-by-step guide that will empower you to tackle similar problems with confidence. Whether you're a student grappling with calculus concepts or a seasoned mathematician seeking a refresher, this exploration will illuminate the intricacies of differentiation.

Delving into the Derivative of g(x) = ln(8 - 5x + 5x³)

The core objective here is to find the derivative, denoted as g'(x), of the function g(x) = ln(8 - 5x + 5x³). This involves applying the chain rule, a cornerstone of calculus that allows us to differentiate composite functions. Composite functions are functions within functions, and in this case, we have the natural logarithm function (ln) acting upon a polynomial function (8 - 5x + 5x³). Understanding and applying the chain rule is paramount to successfully differentiating this type of function.

To effectively find the derivative, we must first grasp the chain rule. This rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In simpler terms, if we have a function h(x) = f(g(x)), then the derivative h'(x) is given by f'(g(x)) * g'(x). This rule provides a systematic way to unravel the layers of a composite function and find its derivative. Applying the chain rule to our function, we'll identify the outer function as the natural logarithm (ln) and the inner function as the polynomial expression (8 - 5x + 5x³).

Now, let's break down the differentiation process step by step. First, we'll identify the outer function, which is the natural logarithm function, ln(u), where u represents the inner function (8 - 5x + 5x³). The derivative of the natural logarithm function, d/du[ln(u)], is 1/u. This is a fundamental derivative that you should have readily available in your calculus toolkit. Next, we'll identify the inner function, which is the polynomial expression u = 8 - 5x + 5x³. The derivative of this polynomial function, du/dx, can be found by applying the power rule to each term. The power rule states that the derivative of x^n is nx^(n-1). Applying this rule, the derivative of 8 is 0, the derivative of -5x is -5, and the derivative of 5x³ is 15x². Therefore, du/dx = -5 + 15x².

With the derivatives of both the outer and inner functions in hand, we can now apply the chain rule. According to the chain rule, g'(x) = (1/u) * (du/dx). Substituting the expressions we found earlier, we get g'(x) = (1/(8 - 5x + 5x³)) * (-5 + 15x²). This is the derivative of the function g(x) = ln(8 - 5x + 5x³). To present the derivative in a more concise form, we can multiply the terms together, resulting in g'(x) = (-5 + 15x²)/(8 - 5x + 5x³). This is the final expression for the derivative of the given function.

A Deep Dive into the Differentiation Process

Let's embark on a comprehensive exploration of the differentiation process, dissecting each step to ensure a thorough understanding. Our mission is to determine the derivative, denoted as g'(x), of the function g(x) = ln(8 - 5x + 5x³). This journey will involve a meticulous application of the chain rule, a cornerstone of calculus that empowers us to differentiate composite functions. Composite functions, as the name suggests, are functions nestled within functions, and in our case, we encounter the natural logarithm function (ln) acting upon a polynomial function (8 - 5x + 5x³). Mastery of the chain rule is paramount to successfully navigate the intricacies of this differentiation problem.

The chain rule, in essence, provides a roadmap for differentiating composite functions. It asserts that the derivative of a composite function is the derivative of the outer function, evaluated at the inner function, meticulously multiplied by the derivative of the inner function. To put it more succinctly, if we have a function h(x) = f(g(x)), then the derivative h'(x) is elegantly expressed as f'(g(x)) * g'(x). This rule offers a systematic approach to unravel the layers of a composite function and unveil its derivative. When we apply the chain rule to our function, we'll pinpoint the outer function as the natural logarithm (ln) and the inner function as the polynomial expression (8 - 5x + 5x³).

Now, let's meticulously dissect the differentiation process, step by step, to ensure clarity and precision. Our initial step involves identifying the outer function, which, in this case, is the natural logarithm function, denoted as ln(u), where u gracefully represents the inner function (8 - 5x + 5x³). The derivative of the natural logarithm function, d/du[ln(u)], is a fundamental result in calculus, elegantly expressed as 1/u. This derivative should be a familiar fixture in your calculus toolkit. Subsequently, we turn our attention to the inner function, which is the polynomial expression u = 8 - 5x + 5x³. The derivative of this polynomial function, du/dx, can be meticulously found by applying the power rule to each term. The power rule, a cornerstone of differentiation, states that the derivative of x^n is gracefully given by nx^(n-1). Applying this rule with precision, the derivative of 8 is 0, the derivative of -5x is -5, and the derivative of 5x³ is elegantly 15x². Therefore, du/dx is concisely expressed as -5 + 15x².

With the derivatives of both the outer and inner functions elegantly determined, we are now poised to apply the chain rule with confidence. According to the chain rule, g'(x) = (1/u) * (du/dx). Substituting the expressions we meticulously derived earlier, we obtain g'(x) = (1/(8 - 5x + 5x³)) * (-5 + 15x²). This is the derivative of the function g(x) = ln(8 - 5x + 5x³). To present the derivative in a more aesthetically pleasing and concise form, we can gracefully multiply the terms together, resulting in g'(x) = (-5 + 15x²)/(8 - 5x + 5x³). This elegant expression represents the final derivative of the given function.

Practical Applications and Further Exploration

Beyond the theoretical realm, understanding derivatives has profound practical applications in various fields. Derivatives are the cornerstone of optimization problems, where we seek to find the maximum or minimum values of a function. They are also crucial in analyzing rates of change, such as velocity and acceleration in physics. In economics, derivatives help determine marginal cost and marginal revenue, aiding in business decision-making. This highlights the far-reaching impact of calculus in shaping our understanding of the world.

For those eager to delve deeper into the world of calculus, numerous resources are available. Textbooks, online courses, and practice problems offer a wealth of knowledge and opportunities to hone your skills. Exploring different types of functions, such as trigonometric functions and exponential functions, will broaden your calculus horizons. Furthermore, understanding the relationship between derivatives and integrals, the other fundamental concept in calculus, will provide a holistic view of this powerful mathematical tool. The journey of calculus is a continuous exploration, and each step forward unlocks new insights and capabilities.

Conclusion: Mastering the Art of Differentiation

In this comprehensive exploration, we have successfully navigated the process of finding the derivative of g(x) = ln(8 - 5x + 5x³). By meticulously applying the chain rule and understanding the derivatives of basic functions, we arrived at the solution: g'(x) = (-5 + 15x²)/(8 - 5x + 5x³). This journey underscores the importance of mastering fundamental calculus concepts, as they form the bedrock for solving more complex problems. Differentiation, as we have seen, is not merely a mathematical exercise; it is a powerful tool that unlocks insights into the behavior of functions and their applications in the real world. Embrace the challenge, delve deeper into the world of calculus, and you will be rewarded with a profound understanding of the mathematical principles that govern our universe.