Derivative Of X^5 / Ln(x^3) A Step-by-Step Solution

Jul 9, 2025 by ADMIN 52 views

Finding the Derivative of y = x^5 / ln(x^3) : A Step-by-Step Guide

In the realm of calculus, finding derivatives is a fundamental operation. Derivatives represent the instantaneous rate of change of a function, providing valuable insights into its behavior. This article will guide you through the process of finding the derivative of the expression y = x^5 / ln(x^3). We'll break down each step, explaining the rules and techniques involved, ensuring you grasp the underlying concepts.

Understanding the Problem

Before diving into the solution, let's clearly define the problem. We are given a function y that depends on x, and our goal is to determine how y changes as x changes. Mathematically, this is represented by the derivative of y with respect to x, denoted as dy/dx or y'. The given function is a quotient, a ratio of two expressions involving x: x^5 in the numerator and ln(x^3) in the denominator. This form suggests that we'll need to employ the quotient rule, a crucial tool in differentiation. The quotient rule is essential when dealing with functions that are expressed as fractions, where both the numerator and denominator are functions of the variable with respect to which we are differentiating. It provides a systematic way to find the derivative of such functions by considering the derivatives of both the numerator and the denominator, along with the original functions themselves. By applying the quotient rule correctly, we can accurately determine the rate of change of the overall function, taking into account the interplay between the changing numerator and denominator. In this particular problem, the quotient rule will enable us to effectively handle the complex structure of the function and arrive at the correct derivative. Furthermore, understanding the quotient rule is not just about memorizing a formula; it is about comprehending the underlying principle of how rates of change interact in a fraction. The quotient rule is not just a standalone formula but a building block for solving more complex differentiation problems. Its application extends beyond simple fractions to scenarios where functions are intertwined in intricate ways. Therefore, mastering the quotient rule is a crucial step in developing a solid foundation in differential calculus and the ability to tackle a wide range of problems involving rates of change and optimization.

Essential Differentiation Rules

To tackle this problem effectively, we need to have a firm grasp of two fundamental differentiation rules: the quotient rule and the chain rule. Let's first look at the quotient rule. The quotient rule is specifically designed for finding the derivative of a function that is expressed as a quotient, meaning a fraction where both the numerator and the denominator are functions of x. Mathematically, if we have a function y = u(x) / v(x), where u(x) and v(x) are differentiable functions, then the quotient rule states that the derivative of y with respect to x is given by: dy/dx = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2. In simpler terms, the derivative of the quotient is found by taking the denominator times the derivative of the numerator, subtracting the numerator times the derivative of the denominator, and then dividing the entire expression by the square of the denominator. The quotient rule is a cornerstone of calculus, essential for handling rational functions and various other complex expressions. Understanding its derivation and application is vital for mastering differentiation. It's not just about memorizing a formula but comprehending how the rates of change of the numerator and denominator interact to influence the rate of change of the overall function. Ignoring the quotient rule can lead to significant errors when differentiating rational functions, so it's crucial to apply it correctly and consistently. Now, let's explore the chain rule. The chain rule is another indispensable tool in calculus, particularly useful when dealing with composite functions, meaning functions within functions. If we have a function y = f(g(x)), where f and g are differentiable functions, then the chain rule states that the derivative of y with respect to x is given by: dy/dx = f'(g(x)) * g'(x). This means we take the derivative of the outer function f evaluated at the inner function g(x), and then multiply it by the derivative of the inner function g'(x). The chain rule allows us to break down complex functions into simpler parts, making differentiation more manageable. It's essential for handling functions like trigonometric functions of polynomials, exponential functions of other functions, and, in our case, the natural logarithm of a polynomial. Mastering the chain rule unlocks the ability to differentiate a vast array of composite functions. Its applications extend far beyond basic calculus, playing a crucial role in fields like physics, engineering, and economics, where composite functions frequently appear in models and equations.

Step-by-Step Solution

Now, let's apply these rules to find the derivative of our given expression, y = x^5 / ln(x^3). This breakdown will provide a structured approach to solving the problem. First, we must identify the u(x) and v(x) components. In our case, we can clearly see that the numerator, x^5, corresponds to u(x), while the denominator, ln(x^3), corresponds to v(x). This identification step is crucial because it sets the stage for applying the quotient rule correctly. Misidentifying these components can lead to incorrect application of the rule and, consequently, a wrong derivative. Therefore, taking the time to carefully analyze the given function and identify its numerator and denominator is a fundamental first step in the differentiation process. Next, we need to find the derivatives of u(x) and v(x) individually. This involves applying basic differentiation rules to each component. For u(x) = x^5, we can use the power rule, which states that the derivative of x^n is n*x^(n-1). Applying this rule, we get u'(x) = 5x^4. For v(x) = ln(x^3), we need to use the chain rule. The derivative of ln(u) is (1/u) * u', where u in this case is x^3. So, the derivative of ln(x^3) is (1/x^3) * (3x^2), which simplifies to 3/x. Understanding these individual derivatives is critical because they are the building blocks for the quotient rule. Any error in finding u'(x) or v'(x) will propagate through the rest of the calculation, leading to an incorrect final answer. Therefore, it's important to double-check these derivatives and ensure their accuracy before proceeding. Now, we apply the quotient rule. Recall that the quotient rule states: dy/dx = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]^2. We now have all the pieces to plug into this formula. Substituting the values we found, we get: dy/dx = [ln(x^3) * 5x^4 - x^5 * (3/x)] / [ln(x^3)]^2. This step involves a direct substitution of the derivatives and original functions into the quotient rule formula. It's a critical step where accuracy is paramount. Any mistake in substitution will lead to an incorrect final answer. Therefore, it's essential to double-check that each term is placed correctly in the formula and that the correct values are used for each function and its derivative. After applying the rule, the next step is to simplify the expression. This often involves algebraic manipulation to reduce the expression to its simplest form. Looking at the expression we derived in the previous step, we can simplify it as follows: dy/dx = [5x^4 * ln(x^3) - 3x^4] / [ln(x^3)]^2. We can factor out x^4 from the numerator to get: dy/dx = x^4 * [5ln(x^3) - 3] / [ln(x^3)]^2. Simplification is crucial not only for obtaining the most concise form of the derivative but also for making it easier to interpret and use in further calculations. A simplified derivative is easier to analyze for critical points, concavity, and other important features of the original function. Furthermore, in many applications, a simplified expression is necessary for further manipulation or for comparing results with other calculations. The final step is to state the final answer. Based on our simplifications, the derivative of the given expression is: dy/dx = x^4 * [5ln(x^3) - 3] / [ln(x^3)]^2. This is the final result of our calculations, representing the instantaneous rate of change of the function y with respect to x. It's important to present the final answer clearly and concisely, ensuring that it is easily understood and can be readily used for further analysis or applications. In summary, we've methodically broken down the problem, applied relevant differentiation rules, and simplified the result to arrive at the derivative of the given function. This step-by-step approach ensures clarity and minimizes the risk of errors, leading to a correct and meaningful solution. Now, let's state our final answer clearly for easy reference: dy/dx = x^4 * [5ln(x^3) - 3] / [ln(x^3)]^2.

Common Mistakes to Avoid

When finding derivatives, particularly with complex expressions, it's easy to make mistakes. Recognizing these common pitfalls can help you avoid them. One frequent mistake is misapplying the quotient rule. The quotient rule has a specific structure, and reversing the order of terms in the numerator or forgetting to square the denominator will lead to an incorrect result. It's vital to memorize the formula correctly and apply it meticulously, ensuring that each term is in its proper place. Another common error arises from incorrectly applying the chain rule. The chain rule involves differentiating the outer function and then multiplying by the derivative of the inner function. Forgetting to multiply by the derivative of the inner function or misidentifying the inner and outer functions can result in a wrong derivative. Careful attention to the structure of composite functions and a systematic application of the chain rule are essential to avoid this mistake. Incorrectly differentiating ln(x) is another pitfall. The derivative of ln(x) is 1/x, but it's easy to forget this or to misapply it when dealing with more complex logarithmic expressions. Remember the basic derivative and use the chain rule when the argument of the logarithm is not simply x. Finally, algebraic errors during simplification are a common source of mistakes. Even if the differentiation steps are correct, errors in simplifying the expression can lead to a wrong final answer. Careful attention to algebraic manipulations, such as factoring, canceling terms, and combining like terms, is crucial. By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy and confidence in finding derivatives. This involves not just knowing the rules but also understanding their application and being meticulous in the process.

Conclusion

Finding the derivative of y = x^5 / ln(x^3) demonstrates the application of fundamental calculus rules. By systematically applying the quotient rule and chain rule, we successfully determined the derivative. This process highlights the importance of understanding and correctly applying differentiation rules, as well as the need for careful algebraic manipulation. Mastering these techniques is crucial for success in calculus and related fields. Through this step-by-step guide, we've shown how to approach complex differentiation problems with confidence and precision, emphasizing the value of clear understanding and meticulous execution. The result of finding the derivatives is: dy/dx = x^4 * [5ln(x^3) - 3] / [ln(x^3)]^2.