Derivative Of X Cosx Sinx

Decoding the Derivative of x cos(x) sin(x): A Comprehensive Guide

Finding the derivative of a function is a cornerstone of calculus. This article provides a detailed explanation of how to derive the derivative of the function x cos(x) sin(x), a problem that combines the product rule and trigonometric identities. We will break down the process step-by-step, clarifying the underlying principles and offering insights beyond a simple solution. Understanding this process will solidify your grasp of differentiation techniques and their application to complex functions.

Introduction: Understanding the Problem

Our goal is to find the derivative of the function f(x) = x cos(x) sin(x). This function involves a mix of algebraic (x) and trigonometric (cos(x) and sin(x)) terms. Because we have multiple functions multiplied together, we need to employ the product rule of differentiation. We'll also leverage trigonometric identities to simplify the final expression and present it in a more elegant form. This problem is commonly encountered in calculus courses and serves as a good exercise in applying fundamental differentiation techniques. The keyword here is "derivative," and related keywords include "calculus," "product rule," "trigonometric identities," and "differentiation."

Step-by-Step Differentiation using the Product Rule

The product rule states that the derivative of a product of two functions is given by:

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

In our case, we can consider the function as a product of three functions: u(x) = x, v(x) = cos(x), and w(x) = sin(x). To apply the product rule to three functions, we'll treat it as a product of two functions: f(x) = [x cos(x)] [sin(x)]. Let's define:

• p(x) = x cos(x)
• q(x) = sin(x)

Then f(x) = p(x)q(x). Now we can apply the product rule:

f'(x) = p'(x)q(x) + p(x)q'(x)

Now we need to find the derivatives of p(x) and q(x). For p(x), we again need to use the product rule since it's a product of x and cos(x):

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

The derivative of q(x) is straightforward:

q'(x) = d/dx [sin(x)] = cos(x)

Now, substitute p'(x), p(x), q'(x), and q(x) back into the expression for f'(x):

f'(x) = [cos(x) - x sin(x)]sin(x) + [x cos(x)]cos(x)

Simplifying the Expression using Trigonometric Identities

Our derivative currently looks rather complex. We can simplify it significantly using trigonometric identities. Let's expand and rearrange the terms:

f'(x) = cos(x)sin(x) - x sin²(x) + x cos²(x)

Notice the presence of sin²(x) and cos²(x). We can utilize the Pythagorean identity:

sin²(x) + cos²(x) = 1

Rearrange this to get: cos²(x) = 1 - sin²(x)

Substituting this into our derivative:

f'(x) = cos(x)sin(x) - x sin²(x) + x(1 - sin²(x))

f'(x) = cos(x)sin(x) - x sin²(x) + x - x sin²(x)

f'(x) = cos(x)sin(x) + x - 2x sin²(x)

We can further simplify using the double angle identity:

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

Substituting this:

f'(x) = sin(2x)/2 + x - 2x sin²(x)

This is a much more concise and elegant representation of the derivative.

Alternative Approach: Using the Product Rule Directly on Three Functions

While the previous method is clear and methodical, we can also apply the product rule directly to the three functions, although it is more complex. Let's denote:

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

The derivative of a product of three functions u(x), v(x), and w(x) is given by:

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

Applying this to our function:

f'(x) = (1)cos(x)sin(x) + x(-sin(x))sin(x) + x cos(x)(cos(x))

f'(x) = cos(x)sin(x) - x sin²(x) + x cos²(x)

Notice that this leads us back to the same expression we obtained earlier, demonstrating the equivalence of both approaches. The simplification process using trigonometric identities remains the same.

Explanation of the Steps and Underlying Concepts

The solution highlights the importance of several key concepts:

• The Product Rule: This is fundamental to differentiating functions that are products of simpler functions. Understanding its application is crucial for success in calculus.
• Trigonometric Identities: These identities are essential for simplifying trigonometric expressions and expressing the derivative in a concise and manageable form. Familiarity with key identities, such as the Pythagorean identity and double-angle identities, is crucial for simplifying complex expressions.
• Chain Rule (implicitly used): Although not explicitly stated, the chain rule is implicitly used when we differentiate sin²(x) and cos²(x), where the outer function is the square and the inner function is the trigonometric function.
• Systematic Approach: A systematic approach to differentiating complex functions is vital. Breaking down the problem into smaller, manageable steps makes the process easier to understand and less prone to errors.

Frequently Asked Questions (FAQ)

• Q: Can this derivative be further simplified? A: While the final form (sin(2x)/2 + x - 2x sin²(x)) is relatively simplified, further simplification depends on the context. There's no single "most simplified" form.

• Q: Why is the product rule necessary here? A: The product rule is necessary because the function is a product of three individual functions (x, cos(x), and sin(x)). Simple term-by-term differentiation is incorrect in this situation.

• Q: What are the applications of this type of derivative? A: Derivatives of functions like this appear in various fields, including physics (modeling oscillations, wave phenomena), engineering (signal processing), and economics (rate of change of complex economic models).

• Q: What if I made a mistake in applying the product rule? A: Carefully review the steps of the product rule. Double-check your differentiation of each individual function. Practice with similar examples to improve your understanding and accuracy.

Conclusion: Mastering Differentiation Techniques

This comprehensive guide demonstrates how to find the derivative of x cos(x) sin(x) by applying the product rule and simplifying the result using trigonometric identities. The process involves a systematic approach, emphasizing the importance of understanding fundamental calculus concepts. The ability to efficiently and accurately differentiate complex functions is a critical skill in various scientific and engineering fields. Remember to practice regularly to solidify your understanding and build confidence in applying these techniques. Through consistent practice and understanding of the underlying principles, you will not only master this specific problem but also develop the skills necessary to tackle more complex derivative problems. The journey of mastering calculus is one of continuous learning and application, so keep practicing and exploring!