Solving For Dy/dx Given Y = Tan X - (1/3) Log X - 2/x A Calculus Guide

Calculus, a fundamental branch of mathematics, deals with the study of continuous change. Within calculus, differentiation is a crucial operation that allows us to determine the rate at which a function's output changes with respect to its input. This article delves into the process of differentiating functions involving trigonometric and logarithmic components, providing a comprehensive guide for students and enthusiasts alike. Specifically, we will explore the differentiation of the function y = tan x - (1/3) log x - 2/x, a common type of problem encountered in calculus courses. By the end of this guide, you will have a solid understanding of how to approach such problems systematically and accurately.

Problem Statement: Finding dy/dx

The core of this exploration lies in finding the derivative of the function y = tan x - (1/3) log x - 2/x with respect to x, denoted as dy/dx. This derivative represents the instantaneous rate of change of y as x varies. To find dy/dx, we will apply several fundamental rules of differentiation, including the derivatives of trigonometric functions, logarithmic functions, and power functions. This problem is a classic example of how different types of functions can be combined, requiring a multifaceted approach to solve. Understanding this process not only aids in solving similar problems but also builds a strong foundation for more advanced topics in calculus. Before we dive into the solution, let's briefly review some essential differentiation rules that will be used throughout this article.

Essential Differentiation Rules

To successfully differentiate the given function, we must be familiar with some fundamental rules. These rules serve as the building blocks for differentiating more complex functions. First and foremost, the derivative of tan x with respect to x is sec^2 x. This is a cornerstone of trigonometric differentiation. Next, we need to recall that the derivative of log x (natural logarithm) with respect to x is 1/x. This rule is essential for handling logarithmic terms. Lastly, we will use the power rule, which states that the derivative of x^n with respect to x is nx^(n-1), where n is any real number. These three rules, when combined with the constant multiple rule and the sum/difference rule, will enable us to tackle the problem at hand. The constant multiple rule states that the derivative of cf(x) is c*f'(x), where c is a constant. The sum/difference rule states that the derivative of f(x) ± g(x) is f'(x) ± g'(x). With these rules in our arsenal, we are well-equipped to find the derivative of the given function.

Step-by-Step Solution

Breaking Down the Function

The given function, y = tan x - (1/3) log x - 2/x, comprises three distinct terms: tan x, (1/3) log x, and 2/x. To differentiate this function, we will apply the sum/difference rule, which allows us to differentiate each term separately and then combine the results. This approach simplifies the process and makes it easier to manage each component of the function individually. By breaking down the function into its constituent parts, we can systematically apply the appropriate differentiation rules to each term. This strategy is particularly useful for functions that involve a mix of trigonometric, logarithmic, and algebraic expressions. Now, let's proceed to differentiate each term one by one, starting with the trigonometric component.

Differentiating tan x

The derivative of tan x with respect to x is a standard result in calculus. The derivative of the tangent function, denoted as d(tan x)/dx, is sec^2 x. This result is derived using the quotient rule on sin x / cos x or can be memorized as one of the fundamental trigonometric derivatives. Understanding this derivative is crucial, as it appears frequently in various calculus problems. The secant function, sec x, is the reciprocal of the cosine function, i.e., sec x = 1/cos x. Therefore, sec^2 x represents the square of the reciprocal of cos x. This derivative, sec^2 x, will be a key component of our final answer. Now that we have addressed the trigonometric term, let's move on to the logarithmic term and determine its derivative.

Differentiating (1/3) log x

The second term in the function is (1/3) log x. To differentiate this term, we will use the constant multiple rule and the derivative of the natural logarithm function. The constant multiple rule states that the derivative of cf(x) is c*f'(x), where c is a constant. In this case, c is 1/3, and f(x) is log x. The derivative of log x with respect to x is 1/x. Therefore, the derivative of (1/3) log x is (1/3) * (1/x), which simplifies to 1/(3x). This result is a direct application of the constant multiple rule and the derivative of the natural logarithm. Understanding how to handle constant multiples in differentiation is essential for simplifying complex expressions. Now that we have differentiated the logarithmic term, let's turn our attention to the final term, which involves a power of x.

Differentiating 2/x

The third term in the function is 2/x, which can be rewritten as 2x^(-1). To differentiate this term, we will use the power rule. The power rule states that the derivative of x^n with respect to x is nx^(n-1), where n is any real number. In this case, we have 2x^(-1), so n is -1. Applying the power rule, we get d(2x^(-1))/dx = 2 * (-1) * x^(-1-1) = -2x^(-2). This can be further simplified to -2/x^2. However, since our original term was -2/x, the derivative of -2/x is -(-2/x^2), which equals 2/x^2. This step is crucial, as it involves both the power rule and careful attention to the negative sign. Now that we have differentiated all three terms, we can combine the results to find the complete derivative dy/dx.

Combining the Derivatives

Having differentiated each term individually, we now combine the results to find dy/dx. We found that the derivative of tan x is sec^2 x, the derivative of (1/3) log x is 1/(3x), and the derivative of -2/x is 2/x^2. Applying the sum/difference rule, we add these derivatives together. Therefore, dy/dx = sec^2 x - 1/(3x) + 2/x^2. This is the final derivative of the given function. This step demonstrates the power of breaking down a complex problem into smaller, manageable parts. By differentiating each term separately and then combining the results, we have successfully found the derivative of the entire function. Now, let's compare our result with the given options to identify the correct answer.

Identifying the Correct Option

Comparing our result, dy/dx = sec^2 x - 1/(3x) + 2/x^2, with the given options, we can identify the correct answer. **The options are:

(a) sec x - 1/(6x) + 2/x^2 (b) sec^2 x - 1/(3x) + 2/x^2 (c) sec x - 3x + x^2 (d) None of these**

Clearly, option (b) matches our derived result. Therefore, the correct answer is (b) sec^2 x - 1/(3x) + 2/x^2. This step is crucial in problem-solving, as it ensures that the derived solution aligns with the given choices. By carefully comparing the results, we can confidently select the correct option. This concludes the step-by-step solution to the problem. In the next section, we will summarize the key steps and reinforce the concepts learned.

Conclusion

In this article, we have walked through the process of differentiating the function y = tan x - (1/3) log x - 2/x. We applied the fundamental rules of differentiation, including the derivatives of trigonometric and logarithmic functions, as well as the power rule. By breaking down the function into individual terms, differentiating each term separately, and then combining the results, we successfully found dy/dx = sec^2 x - 1/(3x) + 2/x^2. This problem exemplifies how a combination of different differentiation rules can be used to solve complex problems in calculus. Understanding these concepts is essential for mastering calculus and its applications in various fields. The key takeaways from this article include the importance of knowing basic differentiation rules, the strategy of breaking down complex functions, and the careful application of each rule. By following these steps, you can confidently tackle similar problems and further enhance your understanding of calculus. This comprehensive guide aims to provide students and enthusiasts with a clear and concise method for differentiating functions involving trigonometric and logarithmic components. Understanding and applying these techniques will not only improve your problem-solving skills but also deepen your appreciation for the elegance and power of calculus.