Finding The Derivative Of F(x) = (10x + 9)^4 / (8x + 5) A Calculus Guide

Introduction

In the realm of calculus, finding the derivative of a function is a fundamental operation. The derivative, denoted as f'(x), provides invaluable insights into the function's behavior, such as its rate of change and critical points. When faced with complex functions, such as the one presented here, $f(x) = (10x + 9)^4 / (8x + 5)$, a systematic approach is crucial to unravel its derivative. This article will serve as a comprehensive guide, meticulously dissecting the function and employing the quotient rule and chain rule to arrive at the derivative, $f'(x)$.

The function at hand, $f(x) = (10x + 9)^4 / (8x + 5)$, is a quotient of two expressions, each involving the variable x. The numerator, $(10x + 9)^4$, is a composite function, where a linear expression is raised to the fourth power. The denominator, $(8x + 5)$, is a simple linear expression. To tackle this complex function, we will strategically employ the quotient rule, which is specifically designed for differentiating functions expressed as a quotient. This rule provides a framework for handling the interplay between the numerator and denominator during differentiation. In addition to the quotient rule, the chain rule will come into play when dealing with the composite function in the numerator. The chain rule allows us to differentiate a function within a function, ensuring that we account for the inner workings of the composite structure. By carefully applying these rules and breaking down the problem into manageable steps, we will systematically navigate the complexities of this function and arrive at its derivative.

Understanding the derivative of a function is not merely an academic exercise; it has profound implications in various fields. In physics, derivatives are used to describe velocity and acceleration, providing insights into the motion of objects. In economics, derivatives help us analyze marginal cost and revenue, aiding in decision-making processes. In engineering, derivatives are crucial for optimizing designs and predicting system behavior. The ability to find the derivative of a complex function like the one presented here is a testament to the power of calculus and its wide-ranging applications. By mastering the techniques outlined in this article, readers will not only be able to solve this specific problem but also gain a deeper understanding of the fundamental principles of calculus and their relevance in the real world.

Applying the Quotient Rule

The quotient rule is a cornerstone of differential calculus, providing a systematic method for finding the derivative of a function that is expressed as the quotient of two other functions. In mathematical terms, if we have a function $f(x)$ defined as $f(x) = u(x) / v(x)$, where $u(x)$ and $v(x)$ are differentiable functions, then the derivative of $f(x)$, denoted as $f'(x)$, can be calculated using the following formula:

$f'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2$

This formula might seem daunting at first, but it is a powerful tool that simplifies the differentiation process for quotient functions. To effectively apply the quotient rule, it is essential to identify the functions $u(x)$ and $v(x)$ correctly and then find their respective derivatives, $u'(x)$ and $v'(x)$. Once these components are determined, plugging them into the formula becomes a straightforward process. The quotient rule ensures that we account for the interplay between the numerator and denominator when finding the derivative, capturing the nuances of how the quotient function changes with respect to the variable x.

In our specific case, we have the function $f(x) = (10x + 9)^4 / (8x + 5)$. To apply the quotient rule, we first identify the numerator and denominator as separate functions. Let's define $u(x) = (10x + 9)^4$ as the numerator and $v(x) = (8x + 5)$ as the denominator. Now that we have identified $u(x)$ and $v(x)$, the next step is to find their respective derivatives, $u'(x)$ and $v'(x)$. Finding $u'(x)$ will require the application of the chain rule, which we will discuss in the next section. For now, let's focus on finding $v'(x)$. Since $v(x) = (8x + 5)$, its derivative is simply the coefficient of x, which is 8. Therefore, $v'(x) = 8$. With $v(x)$ and $v'(x)$ determined, we can now move on to finding $u'(x)$, which will involve the chain rule due to the composite nature of the numerator function.

The quotient rule is not just a formula to be memorized; it is a powerful concept that stems from the fundamental principles of calculus. It is a direct consequence of the limit definition of the derivative and provides a rigorous way to handle the differentiation of quotient functions. Understanding the quotient rule is crucial for anyone working with calculus, as it arises frequently in various applications. By mastering this rule, we equip ourselves with a powerful tool for tackling a wide range of differentiation problems, paving the way for deeper insights into the behavior of functions and their applications in various fields.

Leveraging the Chain Rule

The chain rule is another essential tool in differential calculus, specifically designed for finding the derivative of composite functions. A composite function is a function that is formed by applying one function to the result of another function. In simpler terms, it's a function within a function. The chain rule provides a systematic way to unravel these nested functions and find their derivatives. Mathematically, if we have a composite function $y = f(g(x))$, where $f$ and $g$ are differentiable functions, then the derivative of $y$ with respect to $x$, denoted as $dy/dx$, can be calculated using the following formula:

$dy/dx = f'(g(x)) * g'(x)$

This formula essentially states that the derivative of the composite function is the product of the derivative of the outer function (evaluated at the inner function) and the derivative of the inner function. The chain rule ensures that we account for the contributions of both the outer and inner functions to the overall rate of change of the composite function. To effectively apply the chain rule, it is crucial to identify the outer and inner functions correctly and then find their respective derivatives. Once these components are determined, plugging them into the formula becomes a straightforward process. The chain rule is a powerful tool that extends the scope of differentiation, allowing us to handle complex functions that would otherwise be difficult to differentiate directly.

In our case, when applying the quotient rule to $f(x) = (10x + 9)^4 / (8x + 5)$, we identified the numerator as $u(x) = (10x + 9)^4$. This function is a composite function, where the outer function is raising something to the fourth power, and the inner function is the linear expression $(10x + 9)$. To find the derivative of $u(x)$, denoted as $u'(x)$, we need to employ the chain rule. Let's break down the function further. Let $f(u) = u^4$ be the outer function and $g(x) = 10x + 9$ be the inner function. Now, we need to find the derivatives of these individual functions. The derivative of $f(u) = u^4$ with respect to $u$ is $f'(u) = 4u^3$, using the power rule. The derivative of $g(x) = 10x + 9$ with respect to $x$ is $g'(x) = 10$, as the derivative of a linear expression is simply its coefficient. Now that we have the derivatives of the outer and inner functions, we can apply the chain rule formula.

Applying the chain rule, we have $u'(x) = f'(g(x)) * g'(x)$. Substituting the derivatives we found, we get $u'(x) = 4(10x + 9)^3 * 10$. Simplifying this expression, we obtain $u'(x) = 40(10x + 9)^3$. This is the derivative of the numerator function. We now have all the necessary components to apply the quotient rule fully. We have $u(x) = (10x + 9)^4$, $u'(x) = 40(10x + 9)^3$, $v(x) = (8x + 5)$, and $v'(x) = 8$. Plugging these into the quotient rule formula, we can find the derivative of the original function, $f(x)$. The chain rule is not just a formula to be memorized; it is a powerful concept that stems from the fundamental principles of calculus. It allows us to differentiate complex functions that are built upon other functions, expanding the scope of differentiation significantly. By mastering the chain rule, we equip ourselves with a powerful tool for tackling a wide range of differentiation problems, paving the way for deeper insights into the behavior of composite functions and their applications in various fields.

Synthesizing the Results: Finding f'(x)

Now that we have meticulously applied the quotient rule and the chain rule, we have all the necessary components to determine the derivative of our original function, $f(x) = (10x + 9)^4 / (8x + 5)$. Recall that we defined $u(x) = (10x + 9)^4$ as the numerator and $v(x) = (8x + 5)$ as the denominator. We also found their respective derivatives: $u'(x) = 40(10x + 9)^3$ and $v'(x) = 8$. With these pieces in place, we can now substitute them into the quotient rule formula:

$f'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2$

Substituting the expressions we found, we get:

$f'(x) = [(8x + 5) * 40(10x + 9)^3 - (10x + 9)^4 * 8] / [(8x + 5)]^2$

This expression is the derivative of the function, but it can be further simplified to make it more manageable and easier to interpret. Simplification often involves factoring out common terms and combining like terms. In this case, we can observe that both terms in the numerator have a common factor of $8(10x + 9)^3$. Factoring this out, we get:

$f'(x) = [8(10x + 9)^3 * (5(8x + 5) - (10x + 9))] / [(8x + 5)]^2$

Now, let's simplify the expression inside the parentheses:

$5(8x + 5) - (10x + 9) = 40x + 25 - 10x - 9 = 30x + 16$

Substituting this back into our expression for $f'(x)$, we get:

$f'(x) = [8(10x + 9)^3 * (30x + 16)] / [(8x + 5)]^2$

We can further simplify by factoring out a 2 from the term $(30x + 16)$, giving us $(15x + 8)$. This leads to:

$f'(x) = [16(10x + 9)^3 * (15x + 8)] / [(8x + 5)]^2$

This is the simplified form of the derivative, $f'(x)$. It represents the instantaneous rate of change of the function $f(x)$ at any given point $x$. The simplified expression is not only more concise but also reveals the structure of the derivative more clearly. It highlights the factors that influence the rate of change and provides a more intuitive understanding of the function's behavior. Finding the derivative is just the first step; simplifying it is crucial for extracting meaningful information and applying it to various contexts.

Conclusion

In this comprehensive exploration, we have successfully navigated the intricacies of finding the derivative of the complex function $f(x) = (10x + 9)^4 / (8x + 5)$. We began by understanding the need for a systematic approach when dealing with such functions. We then identified the quotient rule as the primary tool for differentiating quotient functions and the chain rule as the key to unraveling composite functions. By carefully applying these rules, we meticulously dissected the function and found its derivative. The process involved identifying the numerator and denominator, finding their respective derivatives using the chain rule when necessary, and then substituting these components into the quotient rule formula. Finally, we simplified the resulting expression to arrive at the most concise and interpretable form of the derivative:

$f'(x) = [16(10x + 9)^3 * (15x + 8)] / [(8x + 5)]^2$

This final expression represents the instantaneous rate of change of the function $f(x)$ at any given point $x$. It encapsulates the combined effects of the quotient and composite nature of the original function. The derivative provides valuable insights into the function's behavior, such as its increasing and decreasing intervals, critical points, and concavity. These insights are crucial for various applications, including optimization problems, curve sketching, and modeling real-world phenomena. The ability to find the derivative of such a complex function demonstrates a strong understanding of calculus principles and the power of systematic problem-solving.

The techniques and strategies employed in this article are not limited to this specific example. They can be applied to a wide range of differentiation problems involving quotient and composite functions. The quotient rule and chain rule are fundamental tools in calculus, and mastering them is essential for anyone working with derivatives. By understanding the underlying principles and practicing their application, we can confidently tackle complex differentiation problems and gain a deeper appreciation for the power and elegance of calculus. The journey of finding the derivative is not just about arriving at the final answer; it's about the process of breaking down a complex problem into manageable steps, applying the appropriate rules and techniques, and ultimately gaining a deeper understanding of the function's behavior. This understanding is what empowers us to apply calculus to solve real-world problems and make informed decisions.