Derivative Of Y=sin^(1/x) A Step-by-Step Calculus Guide

The derivative of a function represents its instantaneous rate of change, a fundamental concept in calculus with wide-ranging applications in physics, engineering, economics, and various other fields. In this comprehensive guide, we will delve into the process of finding the derivative of the function y = sin^(1/x). This function, while seemingly simple, requires a careful application of differentiation rules and techniques. We will break down the steps involved, providing a clear and detailed explanation for each stage. Mastering the differentiation of such functions is crucial for understanding more complex calculus problems and their real-world implications.

Understanding the Function

Before we dive into the differentiation process, it's essential to understand the function y = sin^(1/x). This function is a composite function, meaning it's formed by combining two simpler functions. Specifically, it involves the sine function and the reciprocal function. The outer function is the sine function, denoted as sin(u), and the inner function is the reciprocal function, u = 1/x. Composite functions like this are commonly encountered in calculus and require the application of the chain rule for differentiation. The chain rule is a powerful tool that allows us to differentiate composite functions by considering the derivatives of the outer and inner functions separately and then combining them appropriately. Understanding the structure of the function is the first step towards successfully finding its derivative. Recognizing the composite nature of y = sin^(1/x) allows us to strategically apply the chain rule, ensuring that we account for the contributions of both the sine and reciprocal functions to the overall rate of change. This initial analysis sets the stage for a systematic and accurate differentiation process.

Applying the Chain Rule

The chain rule is the cornerstone for differentiating composite functions. It states that the derivative of a composite function, y = f(g(x)), is given by dy/dx = f'(g(x)) * g'(x). This rule essentially breaks down the differentiation process into two steps: first, find the derivative of the outer function, f, with respect to the inner function, g(x); second, find the derivative of the inner function, g(x), with respect to x; and finally, multiply these two derivatives together. In our case, y = sin^(1/x), we identify the outer function as f(u) = sin(u) and the inner function as g(x) = 1/x. The derivative of the outer function, f'(u), is cos(u), a well-known trigonometric derivative. The derivative of the inner function, g'(x), is -1/x^2, which can be derived using the power rule for differentiation. Now, applying the chain rule, we multiply these two derivatives together, substituting 1/x back in for u. This gives us the derivative of y = sin^(1/x) as (cos(1/x)) * (-1/x^2). This step-by-step application of the chain rule ensures that we correctly account for the interconnected rates of change within the composite function. The chain rule is not just a formula; it's a fundamental principle that allows us to dissect complex functions and differentiate them systematically.

Differentiating the Outer Function

To apply the chain rule effectively, we first need to differentiate the outer function. In the case of y = sin^(1/x), the outer function is f(u) = sin(u), where u = 1/x. The derivative of the sine function, sin(u), is a fundamental result in calculus: it's simply cos(u). This derivative is derived from the geometric properties of the sine function and is a cornerstone of trigonometric calculus. Understanding this basic derivative is crucial for differentiating more complex trigonometric functions. In our context, differentiating sin(u) with respect to u gives us cos(u). This means that the rate of change of the sine function at any point is given by the cosine of that point. Now, we need to remember that u is not our final variable; it's an intermediate variable that depends on x. So, we have found the first part of the chain rule: the derivative of the outer function with respect to the inner function. This step highlights the importance of knowing basic derivatives and recognizing how they fit into the chain rule framework. The derivative of the sine function is not just a formula to memorize; it's a key building block for understanding the behavior of trigonometric functions and their derivatives.

Differentiating the Inner Function

Next, we need to differentiate the inner function, which in our case is g(x) = 1/x. This function can be rewritten as x^(-1), which makes it easier to apply the power rule of differentiation. The power rule states that the derivative of x^n with respect to x is nx^(n-1)*. Applying this rule to x^(-1), we get -1 * x^(-2), which simplifies to -1/x^2. This derivative represents the rate of change of the reciprocal function with respect to x. As x increases, 1/x decreases, and the negative sign indicates this inverse relationship. The derivative -1/x^2 is a common result and appears in many calculus problems involving rational functions. Now, we have the second piece of the chain rule: the derivative of the inner function with respect to x. This step demonstrates the versatility of the power rule and how it can be applied to functions beyond simple polynomials. The derivative of the inner function is just as important as the derivative of the outer function in the chain rule process; both contributions are necessary to determine the overall rate of change of the composite function.

Combining the Derivatives

Now that we have differentiated both the outer and inner functions, the final step is to combine these derivatives using the chain rule. Recall that the chain rule states dy/dx = f'(g(x)) * g'(x). We found that f'(u) = cos(u) and g'(x) = -1/x^2. Substituting u = 1/x into f'(u), we get cos(1/x). Now, we multiply this by g'(x), which is -1/x^2. This gives us the derivative of y = sin^(1/x) as (cos(1/x)) * (-1/x^2). This expression represents the instantaneous rate of change of y with respect to x. It tells us how y changes as x changes, taking into account the interplay between the sine and reciprocal functions. The negative sign indicates that as x increases, the rate of change of y is generally negative, but the exact behavior depends on the value of cos(1/x). This final expression is the culmination of the chain rule process, demonstrating how the derivatives of the individual components combine to give the derivative of the composite function. This step is crucial for synthesizing the results of the previous steps and arriving at the complete solution. The chain rule is a powerful tool, but it requires careful application and attention to detail to ensure that all components are correctly accounted for.

Simplifying the Result

The derivative we found, (cos(1/x)) * (-1/x^2), can be further simplified to -cos(1/x) / x^2. This simplified form is often preferred because it's more concise and easier to work with in subsequent calculations or analyses. Simplification is a crucial step in calculus, as it can reveal hidden structures and make the result more amenable to interpretation. In this case, the simplified form clearly shows the inverse square relationship between the derivative and x. As x becomes larger, the derivative approaches zero, indicating that the function becomes less sensitive to changes in x. The cosine term introduces oscillatory behavior, but the 1/x^2 term dominates as x becomes large. Simplification not only makes the result more aesthetically pleasing but also provides valuable insights into the function's behavior. In calculus, it's always a good practice to simplify results as much as possible, as this can often lead to a deeper understanding of the underlying mathematical relationships. The simplified derivative -cos(1/x) / x^2 is the final answer to our problem, representing the rate of change of y = sin^(1/x) with respect to x.

Conclusion

In this comprehensive guide, we have successfully found the derivative of y = sin^(1/x) using the chain rule. We started by understanding the function as a composite of the sine and reciprocal functions. Then, we meticulously applied the chain rule, differentiating the outer function, sin(u), and the inner function, 1/x, separately. We combined these derivatives and simplified the result to obtain the final derivative, -cos(1/x) / x^2. This process demonstrates the power and elegance of the chain rule in differentiating composite functions. The chain rule is a fundamental concept in calculus, and mastering its application is essential for solving a wide range of differentiation problems. By breaking down the problem into smaller steps and carefully applying the rules of calculus, we were able to arrive at a clear and concise solution. This example serves as a valuable illustration of how to approach differentiation problems involving composite functions. The derivative we found provides valuable information about the rate of change of the function and its behavior as x varies. This guide should serve as a useful reference for anyone studying calculus or encountering similar differentiation problems.

By understanding the function, applying the chain rule correctly, and simplifying the result, you can confidently find the derivatives of various composite functions. Practice is key to mastering these techniques, and this guide provides a solid foundation for further exploration in calculus.