Step-by-Step Solution Finding The Derivative Of Y = (x Sin X) / (cos X + Sin X)

In the realm of calculus, derivatives serve as the cornerstone for understanding rates of change and the behavior of functions. This article delves into the intricate process of finding the derivative of a specific trigonometric function: $y = \frac{x \sin x}{\cos x + \sin x}$. We will embark on a detailed exploration, employing the quotient rule, product rule, and chain rule to unravel the complexities of this equation and arrive at the desired derivative, $\frac{dy}{dx}$.

Setting the Stage: Understanding the Function and Necessary Calculus Tools

Before we plunge into the differentiation process, let's first familiarize ourselves with the function at hand. The function $y = \frac{x \sin x}{\cos x + \sin x}$ is a composite function, involving trigonometric functions (sine and cosine) and algebraic functions (x). To find its derivative, we will need to leverage several fundamental rules of calculus.

• The Quotient Rule: This rule is essential for differentiating functions expressed as a ratio of two expressions. If $y = \frac{u}{v}$, where u and v are functions of x, then the quotient rule states that $\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$.
• The Product Rule: When dealing with functions that are products of two expressions, the product rule comes into play. If $y = uv$, where u and v are functions of x, then the product rule dictates that $\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$.
• The Chain Rule: This rule is indispensable when differentiating composite functions, where one function is nested within another. If $y = f(g(x))$, then the chain rule asserts that $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$.

With these essential calculus tools in our arsenal, we are well-prepared to tackle the derivative of our trigonometric function.

Applying the Quotient Rule: A Step-by-Step Differentiation Process

To begin, we recognize that our function $y = \frac{x \sin x}{\cos x + \sin x}$ is in the form of a quotient, where the numerator is $u = x \sin x$ and the denominator is $v = \cos x + \sin x$. Therefore, we will commence by applying the quotient rule.

1. Identify u and v:

   • $$u = x \sin x$$

   • $$v = \cos x + \sin x$$

2. Find the derivatives of u and v:
   • To find $\frac{du}{dx}$, we need to apply the product rule since u is the product of x and $\sin x$. Let $u_1 = x$ and $u_2 = \sin x$. Then, $\frac{du_1}{dx} = 1$ and $\frac{du_2}{dx} = \cos x$. Applying the product rule,

     $$\frac{du}{dx} = u_1 \frac{du_2}{dx} + u_2 \frac{du_1}{dx} = x \cos x + \sin x$$

   • To find $\frac{dv}{dx}$, we differentiate $v = \cos x + \sin x$ term by term:

     $$\frac{dv}{dx} = -\sin x + \cos x$$

3. Apply the Quotient Rule Formula: Now that we have identified u, v, $\frac{du}{dx}$, and $\frac{dv}{dx}$, we can plug them into the quotient rule formula:

   $$\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$$

   Substituting the values, we get:

   $$\frac{dy}{dx} = \frac{(\cos x + \sin x)(x \cos x + \sin x) - (x \sin x)(-\sin x + \cos x)}{(\cos x + \sin x)^2}$$

Simplifying the Expression: Unveiling the Derivative in its Concise Form

The expression we obtained from applying the quotient rule is quite lengthy and complex. To arrive at a more manageable and insightful form of the derivative, we need to simplify it through algebraic manipulations.

1. Expand the numerator: We begin by expanding the products in the numerator:

   $$\frac{dy}{dx} = \frac{x \cos^2 x + \cos x \sin x + x \sin x \cos x + \sin^2 x + x \sin^2 x - x \sin x \cos x}{(\cos x + \sin x)^2}$$

2. Combine Like Terms: Next, we combine the like terms in the numerator:

   $$\frac{dy}{dx} = \frac{x \cos^2 x + \cos x \sin x + \sin^2 x + x \sin^2 x}{(\cos x + \sin x)^2}$$

3. Factor and Simplify: Now, we can factor out an x from the first and last terms in the numerator and use the trigonometric identity $\sin^2 x + \cos^2 x = 1$:

   $$\frac{dy}{dx} = \frac{x(\cos^2 x + \sin^2 x) + \cos x \sin x + \sin^2 x}{(\cos x + \sin x)^2}$$

   $$\frac{dy}{dx} = \frac{x + \cos x \sin x + \sin^2 x}{(\cos x + \sin x)^2}$$

The Result: Unveiling the Derivative of the Trigonometric Function

After meticulously applying the quotient rule, product rule, and chain rule, and simplifying the resulting expression, we arrive at the derivative of the function $y = \frac{x \sin x}{\cos x + \sin x}$:

$$\frac{dy}{dx} = \frac{x + \cos x \sin x + \sin^2 x}{(\cos x + \sin x)^2}$$

This equation represents the instantaneous rate of change of the function y with respect to x. It provides valuable insights into the behavior of the function, such as its increasing and decreasing intervals, critical points, and concavity.

Conclusion: Mastering Differentiation Techniques and Their Applications

This exploration has demonstrated the power and versatility of calculus techniques in finding the derivatives of complex functions. By carefully applying the quotient rule, product rule, and chain rule, we successfully determined the derivative of the trigonometric function $y = \frac{x \sin x}{\cos x + \sin x}$. The resulting derivative, $\frac{dy}{dx} = \frac{x + \cos x \sin x + \sin^2 x}{(\cos x + \sin x)^2}$, provides a comprehensive understanding of the function's behavior and its rate of change.

The process of finding derivatives is not merely a mathematical exercise; it has profound implications in various fields, including physics, engineering, economics, and computer science. Derivatives are instrumental in modeling and analyzing dynamic systems, optimizing processes, and making informed decisions. By mastering differentiation techniques, we unlock a powerful tool for understanding and shaping the world around us.

In summary, this article has provided a step-by-step guide to finding the derivative of a trigonometric function, highlighting the importance of fundamental calculus rules and the significance of simplification techniques. The derived expression offers a deeper understanding of the function's behavior and its applications in diverse fields.