Find Dy/dx Given Tan(x-y) = Y/(1+x^2)

Implicit differentiation is a powerful technique in calculus used to find the derivative of a function that is not explicitly defined in terms of a single variable. This method is particularly useful when dealing with equations where it's difficult or impossible to isolate one variable. In this article, we will delve into the process of finding dy/dx for the equation tan(x-y) = y/(1+x^2) using implicit differentiation. We will break down each step, providing clear explanations and justifications to ensure a comprehensive understanding of the technique. This guide aims to equip you with the skills to tackle similar problems confidently and efficiently.

Understanding Implicit Differentiation

Before we dive into the specific problem, let's establish a solid foundation by understanding the concept of implicit differentiation. In contrast to explicit functions, where one variable is directly expressed in terms of another (e.g., y = f(x)), implicit functions define a relationship between variables without explicitly isolating one. For instance, the equation x^2 + y^2 = 1 represents a circle and defines an implicit relationship between x and y. To find dy/dx in such cases, we differentiate both sides of the equation with respect to x, treating y as a function of x. This is where the chain rule becomes crucial, as we need to account for the derivative of y with respect to x whenever we differentiate a term involving y.

Implicit differentiation is vital in various areas of mathematics and physics, especially when dealing with related rates problems, optimization problems, and equations that are inherently implicit. Mastering this technique opens doors to solving a broader range of calculus problems and understanding more complex relationships between variables. In essence, implicit differentiation allows us to find rates of change even when the function is not explicitly defined, making it an indispensable tool in calculus.

When tackling implicit differentiation problems, it is crucial to remember the chain rule and product rule. The chain rule is applied when differentiating a composite function, such as sin(y) or y^2, where y is a function of x. The product rule is essential when differentiating a product of two functions, such as xy or x^2y. Applying these rules correctly ensures that we account for all the dependencies between variables and obtain the accurate derivative. With a clear understanding of these principles, we can confidently approach implicit differentiation problems and successfully find dy/dx for complex equations.

Step-by-Step Solution for tan(x-y) = y/(1+x^2)

Now, let's tackle the problem at hand: finding dy/dx for the equation tan(x-y) = y/(1+x^2) using implicit differentiation. We will proceed step-by-step, explaining each operation and the underlying principles. This meticulous approach will not only help solve this specific problem but also provide a template for handling similar implicit differentiation challenges.

1. Differentiate Both Sides with Respect to x

The first step in implicit differentiation is to differentiate both sides of the equation with respect to x. This ensures that we maintain the equality while introducing the derivatives we need to find dy/dx. Applying this to our equation, tan(x-y) = y/(1+x^2), we get:

d/dx [tan(x-y)] = d/dx [y/(1+x^2)]

This step sets the stage for applying the chain rule and quotient rule, which are essential for differentiating the respective sides of the equation. The left side involves a composite function, tan(x-y), and the right side is a quotient of two functions, y and (1+x^2). By differentiating both sides, we are effectively accounting for the rate of change of each part of the equation with respect to x, paving the way for isolating dy/dx. This initial step is crucial for correctly applying the rules of differentiation and arriving at the solution.

2. Apply the Chain Rule and Quotient Rule

Next, we apply the chain rule to the left side and the quotient rule to the right side of the equation. The chain rule states that if we have a composite function f(g(x)), its derivative is f'(g(x)) * g'(x). The quotient rule states that the derivative of u/v is (v(du/dx) - u(dv/dx)) / v^2. Applying these rules to our differentiated equation, we have:

Left side: d/dx [tan(x-y)] = sec^2(x-y) * d/dx (x-y) = sec^2(x-y) * (1 - dy/dx)

Right side: d/dx [y/(1+x^2)] = [(1+x^2)(dy/dx) - y(2x)] / (1+x²⁾2

Now, our equation looks like this:

sec^2(x-y) * (1 - dy/dx) = [(1+x^2)(dy/dx) - 2xy] / (1+x²⁾2

Here, we've utilized the derivative of tan(u), which is sec^2(u) * du/dx, and the quotient rule to expand and differentiate each side of the original equation. It is essential to apply these rules accurately, paying close attention to the chain rule when differentiating terms involving y. This expansion is a critical step in isolating dy/dx, bringing us closer to the final solution.

3. Simplify and Isolate dy/dx

The next step involves simplifying the equation and isolating dy/dx. This usually requires algebraic manipulation, such as expanding terms, combining like terms, and moving all terms containing dy/dx to one side of the equation. Starting from our equation:

sec^2(x-y) * (1 - dy/dx) = [(1+x^2)(dy/dx) - 2xy] / (1+x²⁾2

First, let's distribute sec^2(x-y) on the left side:

sec^2(x-y) - sec^2(x-y) * (dy/dx) = [(1+x^2)(dy/dx) - 2xy] / (1+x²⁾2

Now, multiply both sides by (1+x²⁾2 to eliminate the denominator:

sec^2(x-y) * (1+x²⁾2 - sec^2(x-y) * (1+x²⁾2 * (dy/dx) = (1+x^2)(dy/dx) - 2xy

Collect terms containing dy/dx on one side and other terms on the other side:

sec^2(x-y) * (1+x²⁾2 + 2xy = (1+x^2)(dy/dx) + sec^2(x-y) * (1+x²⁾2 * (dy/dx)

Factor out dy/dx from the right side:

sec^2(x-y) * (1+x²⁾2 + 2xy = dy/dx * [(1+x^2) + sec^2(x-y) * (1+x²⁾2]

Finally, divide both sides by the factor multiplying dy/dx to isolate it:

dy/dx = [sec^2(x-y) * (1+x²⁾2 + 2xy] / [(1+x^2) + sec^2(x-y) * (1+x²⁾2]

This algebraic manipulation is crucial for obtaining an explicit expression for dy/dx. Each step, from distributing terms to factoring out dy/dx, brings us closer to the final solution. The goal is to isolate dy/dx on one side of the equation, expressing it in terms of x and y. This process often requires careful attention to detail and a solid understanding of algebraic principles. Once we have dy/dx isolated, we have successfully found the derivative implicitly.

4. Final Result

Therefore, the final result for dy/dx is:

dy/dx = [sec^2(x-y) * (1+x²⁾2 + 2xy] / [(1+x^2) + sec^2(x-y) * (1+x²⁾2]

This is the derivative of y with respect to x for the given implicit equation tan(x-y) = y/(1+x^2). This expression provides the slope of the tangent line to the curve defined by the equation at any point (x, y). This final result encapsulates the entire process of implicit differentiation, demonstrating how to systematically find the derivative of implicitly defined functions. The ability to obtain such expressions is crucial in various applications, from analyzing the behavior of curves to solving related rates problems. Understanding the significance of this result helps to solidify the concept of implicit differentiation and its practical applications.

Conclusion

In conclusion, finding dy/dx by implicit differentiation for the equation tan(x-y) = y/(1+x^2) involves a series of crucial steps: differentiating both sides, applying the chain and quotient rules, simplifying the equation, and isolating dy/dx. Each step requires careful attention to detail and a solid understanding of calculus principles. This process highlights the power and versatility of implicit differentiation in handling functions that are not explicitly defined. The final result, dy/dx = [sec^2(x-y) * (1+x²⁾2 + 2xy] / [(1+x^2) + sec^2(x-y) * (1+x²⁾2], provides valuable information about the rate of change of y with respect to x and the slope of the tangent line to the curve. Mastering implicit differentiation opens up a wide range of problem-solving capabilities in calculus and related fields, making it an essential technique for students and professionals alike. By practicing and applying these steps, one can confidently tackle various implicit differentiation problems and gain a deeper understanding of calculus.