Derivative Of X Sin 2x

Finding the Derivative of x sin 2x: A Comprehensive Guide

Finding the derivative of functions involving trigonometric functions and polynomials often requires a solid understanding of differentiation rules. This article provides a detailed explanation of how to find the derivative of x sin 2x, covering the underlying principles, step-by-step calculations, and addressing potential questions. We'll explore the application of the product rule and chain rule, ensuring a comprehensive understanding for students of calculus.

Introduction: Understanding the Problem

The function we're tasked with differentiating is f(x) = x sin 2x. This function is a product of two functions: g(x) = x (a simple polynomial) and h(x) = sin 2x (a trigonometric function with a composite argument). To find its derivative, f'(x), we'll need to utilize the product rule and the chain rule. Let's dive into the necessary rules before tackling the problem directly.

Essential Differentiation Rules: Product Rule and Chain Rule

Before proceeding, let's review the crucial differentiation rules we'll employ:

• The Product Rule: If we have a function f(x) = u(x)v(x), where u(x) and v(x) are both differentiable functions, then the derivative f'(x) is given by:

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

• The Chain Rule: If we have a composite function f(x) = g(h(x)), then its derivative is given by:

  f'(x) = g'(h(x)) * h'(x)

In simpler terms: the chain rule states that you differentiate the "outer" function, leaving the "inner" function alone, and then multiply by the derivative of the "inner" function.

Step-by-Step Differentiation of x sin 2x

Now, let's apply these rules to find the derivative of x sin 2x. We'll break down the process into manageable steps:

1. Identify u(x) and v(x): In our function, f(x) = x sin 2x, let's define:

   • u(x) = x
   • v(x) = sin 2x

2. Find the derivatives of u(x) and v(x):

   • u'(x) = d/dx (x) = 1 (The derivative of x with respect to x is 1)

   • To find v'(x), we need the chain rule since v(x) is a composite function. Let's break it down:

     • Let p(x) = sin x and q(x) = 2x. Then v(x) = p(q(x)).
     • p'(x) = cos x (The derivative of sin x is cos x)
     • q'(x) = 2 (The derivative of 2x with respect to x is 2)
     • Applying the chain rule: v'(x) = p'(q(x)) * q'(x) = cos(2x) * 2 = 2cos(2x)

3. Apply the Product Rule: Now, we can substitute the functions and their derivatives into the product rule formula:

   f'(x) = u'(x)v(x) + u(x)v'(x) f'(x) = (1)(sin 2x) + (x)(2cos 2x) f'(x) = sin 2x + 2x cos 2x

Therefore, the derivative of x sin 2x is sin 2x + 2x cos 2x.

A Deeper Look: Understanding the Chain Rule in this Context

The chain rule is crucial for understanding the derivative of sin 2x. Let's break it down further. The function sin 2x is a composite function. We can visualize it as a function within a function:

• The outer function is the sine function: sin(u)
• The inner function is u = 2x

The chain rule dictates that we differentiate the outer function with respect to the inner function, and then multiply by the derivative of the inner function.

1. Differentiate the outer function: The derivative of sin(u) with respect to u is cos(u).

2. Substitute the inner function: Replace u with 2x, giving us cos(2x).

3. Multiply by the derivative of the inner function: The derivative of 2x with respect to x is 2.

4. Combine: Multiplying the results gives us 2cos(2x), which is the derivative of sin 2x.

Practical Applications and Examples

Understanding the derivative of x sin 2x has applications in various fields, including:

• Physics: Describing oscillatory motion, such as the motion of a pendulum or a spring-mass system, often involves functions similar to x sin 2x. The derivative helps determine velocity and acceleration.

• Engineering: Analyzing signals in electrical engineering or mechanical vibrations uses similar trigonometric functions. The derivative provides insights into the rate of change of these signals.

• Economics: Modeling cyclical phenomena, such as seasonal sales patterns, might employ trigonometric functions. The derivative can help analyze trends and growth rates.

Frequently Asked Questions (FAQ)

Q: Can I use other methods to find the derivative of x sin 2x?

A: While the product rule and chain rule are the most straightforward approaches, other methods could be used, but they would likely involve more complex manipulations. For instance, one might attempt to rewrite the function in a different form before applying differentiation rules, but this often increases the complexity. The product rule and chain rule offer the most efficient solution.

Q: What if the function was x sin(ax + b), where 'a' and 'b' are constants?

A: The process is very similar. You would still use the product rule and the chain rule. The derivative of the inner function would become 'a', resulting in a derivative of: sin(ax + b) + ax cos(ax + b).

Q: What are some common mistakes students make when solving problems like this?

A: Some common mistakes include:

• Forgetting the chain rule: Many students forget to multiply by the derivative of the inner function when differentiating the trigonometric term.

• Incorrect application of the product rule: Failing to correctly add the two terms in the product rule is another frequent error.

• Algebraic errors: Simple mistakes in algebra during the simplification of the final expression can lead to incorrect results.

Conclusion

Finding the derivative of x sin 2x effectively demonstrates the application of fundamental calculus rules, the product rule, and the chain rule. Mastering these rules is essential for tackling more complex differentiation problems. Remember, breaking down the problem into smaller, manageable steps and carefully applying the rules will lead you to the correct answer: sin 2x + 2x cos 2x. Practice is key to building confidence and proficiency in calculus. Continue practicing with variations of this problem, changing the coefficients or introducing more complex trigonometric functions to solidify your understanding.