Finding The Derivative Of Y = Arctan(3x - 5) A Calculus Guide
Introduction
In the realm of calculus, finding the derivatives of inverse trigonometric functions is a fundamental skill. This article delves into the process of determining the derivative of the function y = arctan(3x - 5). We'll provide a step-by-step explanation, ensuring a clear understanding of the underlying concepts and techniques. Mastering such derivatives is crucial for various applications in physics, engineering, and other scientific fields. This guide is designed to be both comprehensive and accessible, catering to students and professionals alike. We will explore the necessary theoretical background, the application of relevant differentiation rules, and the final computation of dy/dx. Throughout this exploration, we aim to elucidate the process in a manner that is not only mathematically rigorous but also intuitively understandable. By the end of this guide, you should be well-equipped to tackle similar derivative problems involving inverse trigonometric functions with confidence and accuracy. Our goal is to transform a potentially daunting task into a manageable and even enjoyable exercise in calculus. Let's embark on this journey of mathematical discovery together!
Understanding Inverse Tangent and Its Derivative
Before diving into the specific problem, it's crucial to grasp the essence of the inverse tangent function, denoted as arctan(x) or tan⁻¹(x). This function provides the angle whose tangent is x. In simpler terms, if y = arctan(x), then tan(y) = x. The range of the arctan(x) function is (-π/2, π/2), which is essential to remember when interpreting results. The derivative of the arctan(x) function is a cornerstone concept we will leverage. The fundamental formula states that the derivative of arctan(x) with respect to x is 1 / (1 + x²). This formula is derived using implicit differentiation and trigonometric identities, and it is a key building block for differentiating more complex functions involving the inverse tangent. Understanding this derivative is not merely about memorization; it's about appreciating its origin and how it fits within the broader context of calculus. This section lays the groundwork for the subsequent steps, ensuring that we approach the problem with a solid conceptual foundation. We will refer back to this basic derivative formula as we tackle the given function, y = arctan(3x - 5). The ability to connect the basic derivative with more complex scenarios is a hallmark of proficiency in calculus.
Applying the Chain Rule
The function y = arctan(3x - 5) is a composite function, meaning it's a function within a function. To find its derivative, we employ the chain rule, a fundamental concept in calculus. The chain 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 our case, the outer function is arctan(u), where u is the inner function (3x - 5). We already know that the derivative of arctan(u) with respect to u is 1 / (1 + u²). Now, we need to find the derivative of the inner function, (3x - 5), with respect to x. This derivative is straightforward: it's simply 3. The chain rule then instructs us to multiply these two derivatives together. This process of breaking down a complex function into its constituent parts and applying the chain rule is a powerful technique in calculus. It allows us to differentiate functions that might otherwise seem intractable. By understanding and applying the chain rule correctly, we can systematically find the derivatives of a wide range of composite functions. The next step will involve putting these pieces together to compute the final derivative of y = arctan(3x - 5).
Step-by-Step Differentiation of y = arctan(3x - 5)
Let's now put the pieces together and find dy/dx for y = arctan(3x - 5). First, we identify the outer function as arctan(u) and the inner function as u = 3x - 5. As established earlier, the derivative of arctan(u) with respect to u is 1 / (1 + u²). Substituting u = 3x - 5, we get the derivative of the outer function evaluated at the inner function as 1 / (1 + (3x - 5)²). Next, we find the derivative of the inner function, u = 3x - 5, with respect to x. This derivative is simply 3. Now, applying the chain rule, we multiply these two results together: dy/dx = [1 / (1 + (3x - 5)²)] * 3. This simplifies to dy/dx = 3 / (1 + (3x - 5)²). This is the derivative of the given function. To further simplify the expression, we can expand the denominator: (3x - 5)² = 9x² - 30x + 25. Adding 1 to this gives us 9x² - 30x + 26. Thus, the final simplified derivative is dy/dx = 3 / (9x² - 30x + 26). This step-by-step approach demonstrates the power of the chain rule and the importance of careful algebraic manipulation in calculus. We have successfully found the derivative of the given function.
Final Result and Simplification
After applying the chain rule and simplifying the expression, we arrive at the final derivative of y = arctan(3x - 5). The derivative, dy/dx, is 3 / (9x² - 30x + 26). This result represents the instantaneous rate of change of y with respect to x for the given function. It's a concise and precise expression that encapsulates the derivative. The simplification process, which involved expanding the square in the denominator and combining like terms, is crucial for presenting the derivative in its most understandable form. A simplified result is not only aesthetically pleasing but also easier to work with in subsequent calculations or applications. This final result can be used to analyze the behavior of the function, such as finding critical points, intervals of increase and decrease, and concavity. It also has applications in related rates problems and optimization problems. Thus, finding and simplifying the derivative is a significant step in understanding the function's properties and its behavior. The derivative dy/dx = 3 / (9x² - 30x + 26) is the culmination of our step-by-step analysis, demonstrating the power of calculus in solving such problems.
Conclusion
In conclusion, we have successfully determined the derivative of y = arctan(3x - 5) using the chain rule and algebraic simplification. The process involved understanding the derivative of the inverse tangent function, recognizing the composite nature of the given function, applying the chain rule, and simplifying the resulting expression. The final derivative, dy/dx = 3 / (9x² - 30x + 26), provides valuable information about the rate of change of the function and can be used in various applications of calculus. This exercise highlights the importance of mastering fundamental calculus concepts and techniques. The ability to differentiate composite functions, especially those involving inverse trigonometric functions, is a crucial skill in calculus. By breaking down the problem into smaller, manageable steps, we were able to arrive at the solution in a clear and systematic manner. This approach is applicable to a wide range of derivative problems. We hope this guide has provided a comprehensive and accessible explanation of the process. Remember, practice is key to mastering calculus, so continue to explore and apply these concepts to various problems. The journey through calculus is one of continuous learning and discovery, and each problem solved brings us closer to a deeper understanding of the subject.
