Differentiating Y = Ln(|6 - 5x^7|) A Step-by-Step Guide

Introduction to Differentiation of Logarithmic Functions

In the realm of calculus, differentiating functions is a fundamental operation that allows us to determine the rate at which a function's output changes with respect to its input. This is particularly crucial in various fields, including physics, engineering, economics, and computer science, where understanding rates of change is essential for modeling and analyzing dynamic systems. When it comes to logarithmic functions, differentiation requires a specific set of rules and techniques. Logarithmic functions are the inverses of exponential functions and appear frequently in mathematical models due to their ability to simplify complex relationships and describe phenomena involving growth and decay. Differentiating these functions correctly is vital for solving problems related to optimization, curve sketching, and related rates. In this article, we will delve into the process of differentiating a particular logarithmic function, $y = ln(|6 - 5x^7|)$, providing a step-by-step explanation to ensure clarity and understanding. By mastering this process, you will be better equipped to tackle more complex calculus problems involving logarithmic and absolute value functions. Understanding the nuances of differentiation not only enhances your mathematical toolkit but also sharpens your analytical skills, enabling you to approach real-world problems with greater confidence and precision. Furthermore, differentiating logarithmic functions often involves the application of the chain rule, a cornerstone of differential calculus. This rule allows us to differentiate composite functions, which are functions within functions. In our case, the outer function is the natural logarithm, and the inner function is an absolute value expression involving a polynomial. Correctly applying the chain rule is key to obtaining the correct derivative. The absolute value function also adds an extra layer of complexity, as it requires us to consider cases where the expression inside the absolute value is positive or negative. This detailed consideration is essential for ensuring the derivative is accurate across the entire domain of the function. By working through this example, we will reinforce your understanding of these critical concepts and techniques in differential calculus.

Understanding the Function: y = ln(|6 - 5x^7|)

The function we aim to differentiate is $y = ln(|6 - 5x^7|)$. This function is a composition of several functions, making it an excellent example for illustrating the chain rule in differentiation. To break it down, we have the natural logarithm function, denoted as $ln(u)$, where $u$ is another function. In this case, $u$ is the absolute value function $|6 - 5x^7|$. The absolute value function itself contains a polynomial function, $6 - 5x^7$. Understanding this composition is crucial for applying the chain rule effectively. The natural logarithm, denoted as $ln(x)$, is the logarithm to the base $e$, where $e$ is an irrational number approximately equal to 2.71828. The natural logarithm is the inverse of the exponential function $e^x$. It is widely used in mathematics and its applications due to its convenient properties in calculus. One of the key properties is its derivative: the derivative of $ln(x)$ with respect to $x$ is $1/x$. However, in our case, we have $ln(|6 - 5x^7|)$, which introduces additional complexity due to the absolute value function. The absolute value function, denoted as $|x|$, returns the non-negative value of $x$. In other words, if $x$ is positive or zero, $|x| = x$, and if $x$ is negative, $|x| = -x$. This piecewise nature of the absolute value function means we must consider different cases when differentiating. For the expression $|6 - 5x^7|$, we have to consider when $6 - 5x^7$ is positive or negative. The polynomial function inside the absolute value, $6 - 5x^7$, is a seventh-degree polynomial. Differentiating polynomial functions is relatively straightforward using the power rule, which states that the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$. However, the presence of the absolute value and the natural logarithm means we cannot directly apply the power rule. Instead, we need to use the chain rule in conjunction with the derivative rules for logarithmic and absolute value functions. The chain rule is a fundamental rule in calculus that allows us to differentiate composite functions. It states that if we have a function $y = f(g(x))$, then the derivative of $y$ with respect to $x$ is given by $dy/dx = f'(g(x)) * g'(x)$. In simpler terms, we differentiate the outer function while keeping the inner function intact, then multiply by the derivative of the inner function. Applying the chain rule correctly is essential for differentiating our given function, $y = ln(|6 - 5x^7|)$.

Applying the Chain Rule: Step-by-Step Differentiation

To differentiate $y = ln(|6 - 5x^7|)$, we will apply the chain rule step by step. Let's break down the function into its components. First, we identify the outermost function as the natural logarithm, $ln(u)$, where $u = |6 - 5x^7|$. The derivative of $ln(u)$ with respect to $u$ is $1/u$. Next, we need to consider the derivative of the absolute value function, $|6 - 5x^7|$. The derivative of $|v|$ with respect to $v$ is given by $v/|v|$ when $v e 0$. In our case, $v = 6 - 5x^7$. So, the derivative of $|6 - 5x^7|$ with respect to $x$ will involve the term $(6 - 5x^7)/|6 - 5x^7|$. Now, we need to find the derivative of the innermost function, which is the polynomial $6 - 5x^7$. Applying the power rule, the derivative of $6$ (a constant) is $0$, and the derivative of $-5x^7$ is $-5 * 7x^{7-1} = -35x^6$. Thus, the derivative of $6 - 5x^7$ with respect to $x$ is $-35x^6$. We can now apply the chain rule. The chain rule states that if $y = f(g(h(x)))$, then $y' = f'(g(h(x))) * g'(h(x)) * h'(x)$. In our case, this translates to: y' = rac{d}{dx} ln(|6 - 5x^7|) = rac{1}{|6 - 5x^7|} * rac{6 - 5x^7}{|6 - 5x^7|} * (-35x^6). Simplifying this expression, we get: y' = rac{1}{|6 - 5x^7|} * rac{6 - 5x^7}{|6 - 5x^7|} * (-35x^6). We can further simplify this by canceling out $|6 - 5x^7|$ in the numerator and denominator of the first two terms: y' = rac{1}{6 - 5x^7} * (-35x^6). Thus, the final derivative is: y' = rac{-35x^6}{6 - 5x^7}. This result gives us the rate of change of $y$ with respect to $x$. It's important to note that this derivative is valid for all $x$ such that $6 - 5x^7 e 0$, as the original function is not defined when the absolute value expression is zero. The application of the chain rule in this scenario highlights its power in handling composite functions. By systematically differentiating each layer of the function and multiplying the results, we arrive at the derivative of the entire function. This process is fundamental in calculus and is used extensively in various applications. The derivative we found, y' = rac{-35x^6}{6 - 5x^7}, provides valuable information about the behavior of the original function. For example, it can be used to find critical points, intervals of increasing and decreasing behavior, and points of inflection. Understanding these aspects is crucial for sketching the graph of the function and analyzing its properties.

Simplifying and Analyzing the Resulting Derivative

After applying the chain rule, we obtained the derivative y' = rac{-35x^6}{6 - 5x^7}. This derivative is a rational function, which is a ratio of two polynomials. Analyzing this derivative involves understanding its behavior, including its critical points, intervals of increase and decrease, and any asymptotes. The numerator of the derivative, $-35x^6$, is always non-positive since $x^6$ is always non-negative and the coefficient is negative. This indicates that the derivative will be zero when $x = 0$ and negative for all other $x$ values (except where the denominator is zero). The denominator of the derivative, $6 - 5x^7$, is a polynomial of degree 7. The sign of the denominator will change depending on the value of $x$. To find when the denominator is zero, we solve $6 - 5x^7 = 0$, which gives x^7 = rac{6}{5}, and thus x = (rac{6}{5})^{1/7}. This value of $x$ is a critical point for the derivative, as the derivative is undefined at this point. To analyze the intervals of increase and decrease, we need to consider the sign of the derivative in different intervals. The derivative y' = rac{-35x^6}{6 - 5x^7} will be positive when the numerator and denominator have opposite signs, and negative when they have the same sign. Since the numerator is always non-positive, the sign of the derivative depends on the sign of the denominator. - When x < (rac{6}{5})^{1/7}, $5x^7 < 6$, so $6 - 5x^7 > 0$. Thus, $y' < 0$, meaning the function is decreasing. - When x > (rac{6}{5})^{1/7}, $5x^7 > 6$, so $6 - 5x^7 < 0$. Thus, $y' > 0$, meaning the function is increasing. - At $x = 0$, $y' = 0$, which is a critical point. To determine the nature of this critical point, we can use the first derivative test. Since the derivative is negative to the left of $x = 0$ and negative to the right of $x = 0$, there is no change in sign, indicating that $x = 0$ is neither a local minimum nor a local maximum. It is a stationary point. The asymptotes of the derivative are important for understanding its behavior as $x$ approaches certain values. In this case, there is a vertical asymptote at x = (rac{6}{5})^{1/7}, where the denominator is zero. There are no horizontal asymptotes because the degree of the numerator (6) is less than the degree of the denominator (7). Analyzing the derivative provides a comprehensive understanding of the behavior of the original function. By finding critical points, intervals of increase and decrease, and asymptotes, we can sketch the graph of the function and understand its properties more fully. This analysis is a crucial step in calculus and is used in various applications to model and solve real-world problems.

Conclusion: Key Takeaways and Applications

In this article, we have successfully differentiated the function $y = ln(|6 - 5x^7|)$ using the chain rule. This process involved breaking down the function into its composite parts—the natural logarithm, the absolute value, and the polynomial—and applying the derivative rules for each part sequentially. The resulting derivative, y' = rac{-35x^6}{6 - 5x^7}, provides valuable insights into the behavior of the original function. One of the key takeaways from this exercise is the importance of the chain rule in calculus. The chain rule allows us to differentiate complex composite functions by systematically differentiating each layer of the function and multiplying the results. This technique is fundamental in calculus and is used extensively in various applications. We also highlighted the significance of the absolute value function in differentiation. The absolute value function introduces piecewise behavior, requiring us to consider different cases based on the sign of the expression inside the absolute value. Understanding how to differentiate absolute value functions is crucial for handling functions that exhibit non-differentiable points. Furthermore, the analysis of the derivative is a critical step in understanding the original function's behavior. By finding critical points, intervals of increase and decrease, and asymptotes, we can sketch the graph of the function and understand its properties more fully. This analysis is used in a wide range of applications, including optimization problems, curve sketching, and the study of related rates. The applications of differentiation extend beyond pure mathematics. In physics, derivatives are used to describe velocity and acceleration. In engineering, they are used to optimize designs and model dynamic systems. In economics, they are used to analyze marginal cost and marginal revenue. In computer science, they are used in machine learning algorithms and optimization techniques. Mastering the techniques of differentiation, including the chain rule and the handling of absolute value functions, is essential for anyone working in these fields. The ability to differentiate functions accurately and efficiently is a valuable skill that enables us to solve complex problems and make informed decisions. In conclusion, differentiating $y = ln(|6 - 5x^7|)$ not only provides a specific derivative but also reinforces key concepts in calculus and their broader applications. By understanding these concepts, we can tackle more complex problems and apply these techniques to real-world scenarios.