Derivative Of Y=x Sin X / (cos X + Sin X) A Step-by-Step Guide
In the realm of calculus, finding the derivative of a function is a fundamental operation that reveals the instantaneous rate of change of the function. This article delves into the intricate process of determining 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 and other essential calculus techniques to arrive at the solution. This exploration will not only enhance your understanding of differentiation but also provide a solid foundation for tackling more complex calculus problems.
Understanding the Foundation: Derivatives and Trigonometric Functions
Before diving into the intricacies of the derivative, it's crucial to establish a firm grasp of the fundamental concepts of derivatives and trigonometric functions. Derivatives, at their core, represent the instantaneous rate at which a function's output changes with respect to its input. In simpler terms, the derivative of a function at a specific point provides the slope of the line tangent to the function's graph at that point. This concept has far-reaching applications in physics, engineering, economics, and various other fields, allowing us to model and analyze dynamic systems.
Trigonometric functions, such as sine, cosine, and tangent, are the cornerstone of mathematical modeling of periodic phenomena, including oscillations, waves, and rotations. Their derivatives exhibit unique patterns that are essential to recognize. Specifically, the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). These fundamental relationships form the bedrock of differentiating more complex trigonometric expressions.
The function we aim to differentiate, $y=\frac{x \sin x}{\cos x+\sin x}$, is a combination of algebraic and trigonometric elements. It incorporates the product of x and sin(x) in the numerator and the sum of cos(x) and sin(x) in the denominator. This structure necessitates the application of the quotient rule, a powerful tool for differentiating functions expressed as fractions.
The Quotient Rule: A Key to Unlocking the Derivative
The quotient rule is a cornerstone of differential calculus, providing a systematic approach to finding the derivative of a function that is expressed as the ratio of two other functions. It's an indispensable tool in situations where we encounter fractions involving variables, such as the function $y=\frac{x \sin x}{\cos x+\sin x}$ that we are investigating. The quotient rule states that if we have a function y defined as y = u/v, where both u and v are functions of x, then the derivative of y with respect to x, denoted as dy/dx, is given by:
In this formula:
- u represents the numerator of the original function.
- v represents the denominator of the original function.
- du/dx is the derivative of the numerator u with respect to x.
- dv/dx is the derivative of the denominator v with respect to x.
To effectively apply the quotient rule, we must carefully identify the numerator (u) and the denominator (v) in our function. In our case, for the function $y=\frac{x \sin x}{\cos x+\sin x}$, we can clearly see that:
- u = x sin x
- v = cos x + sin x
Once we have correctly identified u and v, the next step involves finding their respective derivatives, du/dx and dv/dx. This often requires applying other differentiation rules, such as the product rule or the chain rule, depending on the complexity of u and v. In the subsequent sections, we will meticulously calculate du/dx and dv/dx for our specific functions.
The quotient rule is not just a formula to be memorized; it's a powerful technique that allows us to break down complex derivatives into manageable steps. By understanding the underlying logic of the rule and practicing its application, we can confidently tackle a wide range of differentiation problems involving fractional expressions.
Deconstructing the Numerator: Applying the Product Rule
The numerator of our function, u = x sin x, is itself a product of two functions: x and sin x. To find the derivative of u with respect to x, denoted as du/dx, we must employ another fundamental rule of calculus known as the product rule. The product rule provides a systematic way to differentiate functions that are formed by multiplying two other functions together.
The product rule states that if we have a function u defined as u = f(x)g(x), where both f(x) and g(x) are functions of x, then the derivative of u with respect to x, denoted as du/dx, is given by:
In this formula:
- f(x) and g(x) are the two functions being multiplied.
- f'(x) is the derivative of the function f(x) with respect to x.
- g'(x) is the derivative of the function g(x) with respect to x.
To apply the product rule to our numerator, u = x sin x, we can identify the two functions as follows:
- f(x) = x
- g(x) = sin x
Now, we need to find the derivatives of f(x) and g(x) with respect to x:
- f'(x) = \frac{d}{dx}(x) = 1 (The derivative of x with respect to x is simply 1.)
- g'(x) = \frac{d}{dx}(sin x) = cos x (The derivative of sin x with respect to x is cos x.)
Having determined f(x), g(x), f'(x), and g'(x), we can now substitute these values into the product rule formula:
Therefore, the derivative of the numerator, u = x sin x, with respect to x is du/dx = sin x + x cos x. This result will be a crucial component in applying the quotient rule to find the overall derivative of our original function.
Differentiating the Denominator: A Straightforward Approach
Our next task is to determine the derivative of the denominator, v = cos x + sin x, with respect to x, which we denote as dv/dx. This step is relatively straightforward, as it involves differentiating the sum of two trigonometric functions. To find the derivative of a sum of functions, we can simply differentiate each term individually and then add the results.
In our case, we have:
- v = cos x + sin x
To find dv/dx, we need to differentiate both cos x and sin x with respect to x:
- \frac{d}{dx}(cos x) = -sin x (The derivative of cos x with respect to x is -sin x.)
- \frac{d}{dx}(sin x) = cos x (The derivative of sin x with respect to x is cos x.)
Now, we can add these derivatives together to find dv/dx:
Therefore, the derivative of the denominator, v = cos x + sin x, with respect to x is dv/dx = -sin x + cos x. This result, along with the derivative of the numerator we found earlier, will allow us to apply the quotient rule and find the derivative of our original function.
Applying the Quotient Rule: Bringing It All Together
Now that we have meticulously calculated the derivatives of both the numerator (du/dx) and the denominator (dv/dx), we are fully equipped to apply the quotient rule and determine the derivative of our original function, $y=\frac{x \sin x}{\cos x+\sin x}$. Recall that the quotient rule states:
We have the following components:
- u = x sin x
- v = cos x + sin x
-
-
Substituting these values into the quotient rule formula, we get:
This expression represents the derivative of our function, but it can be further simplified to a more elegant form. The next step involves expanding the terms in the numerator and then looking for opportunities to combine like terms and potentially factor the expression.
Simplifying the Expression: A Journey to Elegance
The expression we obtained after applying the quotient rule, $\frac{dy}{dx} = \frac{(cos x + sin x)(sin x + x cos x) - (x sin x)(-sin x + cos x)}{(cos x + sin x)^2}$, is a valid representation of the derivative, but it is not in its simplest form. To achieve a more concise and insightful result, we need to embark on a journey of algebraic simplification.
The first step in this simplification process is to expand the terms in the numerator. This involves carefully multiplying each term within the parentheses and paying close attention to signs. Let's expand the first term in the numerator, (cos x + sin x)(sin x + x cos x):
(cos x + sin x)(sin x + x cos x) = cos x * sin x + cos x * x cos x + sin x * sin x + sin x * x cos x
Simplifying this, we get:
cos x sin x + x cos²x + sin²x + x sin x cos x
Now, let's expand the second term in the numerator, -(x sin x)(-sin x + cos x):
-(x sin x)(-sin x + cos x) = x sin x sin x - x sin x cos x
Simplifying this, we get:
x sin²x - x sin x cos x
Now, let's combine these expanded terms in the numerator:
cos x sin x + x cos²x + sin²x + x sin x cos x + x sin²x - x sin x cos x
Notice that the terms x sin x cos x and - x sin x cos x cancel each other out. This leaves us with:
cos x sin x + x cos²x + sin²x + x sin²x
We can further simplify this by grouping the terms involving x:
cos x sin x + x(cos²x + sin²x) + sin²x
Recall the fundamental trigonometric identity: sin²x + cos²x = 1. Applying this identity, we get:
cos x sin x + x(1) + sin²x = cos x sin x + x + sin²x
Now, let's rewrite the numerator with the simplified form:
The denominator is (cos x + sin x)², let's expand it:
(cos x + sin x)² = cos²x + 2 cos x sin x + sin²x
Again, using the identity sin²x + cos²x = 1, we have:
1 + 2 cos x sin x
Thus, the final simplified form of the derivative is:
While this form is significantly simplified compared to the initial expression obtained from the quotient rule, there might be other equivalent forms depending on the desired level of simplification or the context in which the derivative is being used. However, this simplified form provides a clearer understanding of the derivative's behavior and its relationship to the original function.
Conclusion: Mastering the Art of Differentiation
In this comprehensive exploration, we have successfully navigated the process of finding the derivative of the trigonometric function $y=\frac{x \sin x}{\cos x+\sin x}$. Our journey began with a solid foundation in the concepts of derivatives and trigonometric functions, followed by the strategic application of the quotient rule and the product rule. We meticulously calculated the derivatives of the numerator and denominator, and then skillfully applied the quotient rule formula. The final step involved a careful simplification process, where we expanded terms, combined like terms, and utilized trigonometric identities to arrive at a more elegant expression for the derivative.
This exercise serves as a testament to the power and elegance of calculus in analyzing complex functions. The quotient rule, in particular, is a versatile tool that enables us to differentiate functions expressed as fractions, and the product rule allows us to handle functions formed by the product of other functions. By mastering these techniques, you can confidently tackle a wide range of differentiation problems in calculus and related fields.
The derivative we obtained, $\frac{dy}{dx} = \frac{cos x sin x + x + sin²x}{1 + 2 cos x sin x}$, provides valuable information about the instantaneous rate of change of the original function. It can be used to analyze the function's behavior, find critical points, determine intervals of increase and decrease, and solve optimization problems. The applications of derivatives are vast and far-reaching, making them an indispensable tool in mathematics, physics, engineering, economics, and numerous other disciplines.
As you continue your exploration of calculus, remember that practice is key to mastery. Work through numerous examples, challenge yourself with complex problems, and strive to understand the underlying concepts. With dedication and perseverance, you can unlock the full potential of calculus and its applications.
