Derivative Of X Cosx Sinx

Finding the Derivative of x cos(x) sin(x): A complete walkthrough

Finding the derivative of a function like x cos(x) sin(x) might seem daunting at first, but breaking it down into manageable steps reveals a straightforward process. In real terms, this article will guide you through the calculation, explaining the underlying calculus principles and providing a clear, step-by-step solution. Now, we'll also explore some related concepts and address frequently asked questions. By the end, you'll not only understand how to solve this specific problem but also gain a deeper understanding of differentiation techniques applicable to a wider range of functions.

Introduction: Understanding the Problem

Our goal is to find the derivative of the function f(x) = x cos(x) sin(x). This requires applying several differentiation rules, primarily the product rule and the chain rule. The product rule is essential because we have a product of three functions: x, cos(x), and sin(x). The chain rule, while not directly involved in the primary derivation, underpins the derivatives of the trigonometric functions.

Before diving into the solution, let's refresh our understanding of the key rules:

1. Product Rule: The derivative of a product of two functions, u(x) and v(x), is given by:

d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

2. Chain Rule: The derivative of a composite function, f(g(x)), is given by:

d/dx [f(g(x))] = f'(g(x)) * g'(x)

3. Derivatives of Trigonometric Functions:

• d/dx [sin(x)] = cos(x)
• d/dx [cos(x)] = -sin(x)

Step-by-Step Solution: Differentiating x cos(x) sin(x)

To simplify the calculation, let's first combine the trigonometric functions using a trigonometric identity:

cos(x)sin(x) = (1/2)sin(2x)

Now our function becomes:

f(x) = (1/2)x sin(2x)

This simplifies the application of the product rule. Now we have a product of two functions: u(x) = (1/2)x and v(x) = sin(2x).

Applying the product rule:

f'(x) = u'(x)v(x) + u(x)v'(x)

First, find the derivatives of u(x) and v(x):

u'(x) = d/dx [(1/2)x] = 1/2

v'(x) = d/dx [sin(2x)] Here, we need the chain rule. Let's define g(x) = 2x, then v(x) = sin(g(x)).

v'(x) = cos(2x) * d/dx (2x) = 2cos(2x)

Now, substitute these derivatives into the product rule formula:

f'(x) = (1/2)sin(2x) + (1/2)x * 2cos(2x)

Simplifying:

f'(x) = (1/2)sin(2x) + xcos(2x)

Because of this, the derivative of x cos(x) sin(x) is (1/2)sin(2x) + xcos(2x)

Alternative Approach: Applying the Product Rule Directly

Alternatively, we can apply the product rule directly to the original function, x cos(x) sin(x), although this method is slightly more complex. Let's define:

• u(x) = x
• v(x) = cos(x)
• w(x) = sin(x)

Then, f(x) = u(x)v(x)w(x). We need to use the product rule multiple times. First consider the product u(x)v(x):

d/dx[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) = 1cos(x) + x(-sin(x)) = cos(x) - xsin(x)

Now, let's multiply the result by w(x) and apply the product rule again:

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d/dx[(cos(x) - xsin(x))sin(x)] = (d/dx[cos(x) - xsin(x)])sin(x) + (cos(x) - xsin(x))(d/dx[sin(x)])

Calculating the derivative of (cos(x) - xsin(x)):

d/dx[cos(x) - xsin(x)] = -sin(x) - (sin(x) + xcos(x)) = -2sin(x) - xcos(x)

Substituting back into the equation:

d/dx[xcos(x)sin(x)] = (-2sin(x) - xcos(x))sin(x) + (cos(x) - xsin(x))cos(x)

Expanding and simplifying:

= -2sin²(x) - xsin(x)cos(x) + cos²(x) - xsin(x)cos(x)

= cos²(x) - 2sin²(x) - 2xsin(x)cos(x)

Using trigonometric identities: cos²(x) - sin²(x) = cos(2x) and 2sin(x)cos(x) = sin(2x), we can simplify to:

= cos(2x) - sin²(x) - xsin(2x)

While this approach leads to the same result after simplification using trigonometric identities, the first approach (combining trigonometric functions initially) is more efficient.

Scientific Explanation and Further Exploration

The derivative represents the instantaneous rate of change of the function. In practice, in this case, f'(x) = (1/2)sin(2x) + xcos(2x) tells us how quickly the value of x cos(x) sin(x) is changing at any given point x. The presence of both sine and cosine terms indicates an oscillatory behavior in the rate of change. The x term in the second part contributes to a changing amplitude of this oscillation.

This problem highlights the power of combining algebraic manipulation with calculus rules. That said, choosing the right approach, such as using trigonometric identities to simplify the function before differentiation, significantly reduces the complexity of the calculation. To build on this, understanding the chain rule and product rule is crucial for tackling a wide range of derivative problems, including those involving more complex composite functions.

Frequently Asked Questions (FAQ)

Q1: Why did we use the trigonometric identity cos(x)sin(x) = (1/2)sin(2x)?

A1: This identity simplified the function, reducing the number of terms and making the application of the product rule much easier. It streamlined the calculation and avoided the more complex chain of differentiation involved in the alternative approach.

Q2: Is there another way to solve this problem?

A2: Yes, as shown above, we could apply the product rule directly to the three functions (x, cos(x), sin(x)). Now, this is a valid approach, but it leads to a more complex intermediate expression that requires further simplification using trigonometric identities to reach the final answer. The first method is generally more efficient.

Q3: What are some practical applications of finding derivatives of such functions?

A3: Derivatives are fundamental in various fields. Also, in engineering, they help model rates of change in systems. So in physics, they are used to describe velocities and accelerations. In economics, they are used in optimization problems. The specific function x cos(x) sin(x) might not have a direct physical meaning, but the techniques used to find its derivative are broadly applicable to real-world problems.

Q4: How can I practice solving similar problems?

A4: Practice is key! Start with simpler examples and gradually increase the complexity. Try differentiating similar functions involving products of trigonometric and polynomial functions. Online resources, textbooks, and practice problem sets can provide ample opportunities for practice.

Conclusion

Finding the derivative of x cos(x) sin(x) involves a strategic application of the product rule and a prudent use of trigonometric identities. Here's the thing — the ability to manipulate functions and apply calculus rules effectively is crucial for success in various scientific and technical disciplines. Day to day, the solution, (1/2)sin(2x) + xcos(2x), represents the instantaneous rate of change of the original function. Understanding this process reinforces the importance of mastering differentiation techniques and employing strategic problem-solving skills. In real terms, this article illustrated two approaches, highlighting the advantage of simplifying the function before differentiation whenever possible. Remember, consistent practice and a deeper understanding of underlying principles are key to mastering calculus.