Derivative Of Xcosx + Sinx

Understanding the Derivative of xcos(x) + sin(x): A Comprehensive Guide

Finding the derivative of a function is a fundamental concept in calculus. This article provides a detailed explanation of how to find the derivative of the function xcos(x) + sin(x), covering the underlying principles and techniques involved. We'll explore the necessary rules of differentiation, step-by-step calculations, and delve into the practical applications of this derivative. This comprehensive guide is designed for students of calculus, from beginners to those seeking a deeper understanding of differentiation.

Introduction

The function f(x) = xcos(x) + sin(x) combines polynomial and trigonometric functions. To find its derivative, f'(x), we'll employ the rules of differentiation, specifically the sum rule, the product rule, and the derivatives of trigonometric functions. Understanding these rules is crucial for mastering differentiation techniques. This exploration will go beyond simply providing the answer; we will unpack the methodology and explain the reasoning behind each step, ensuring a clear understanding of the process.

Essential Differentiation Rules

Before we tackle the derivative of xcos(x) + sin(x), let's refresh our memory on the key differentiation rules:

• Sum Rule: The derivative of a sum of functions is the sum of their derivatives. Mathematically, if f(x) = g(x) + h(x), then f'(x) = g'(x) + h'(x).

• Product Rule: The derivative of a product of two functions is given by: if f(x) = g(x)h(x), then f'(x) = g'(x)h(x) + g(x)h'(x).

• Derivatives of Trigonometric Functions:

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

These rules form the foundation for differentiating complex functions.

Step-by-Step Differentiation of xcos(x) + sin(x)

Now, let's apply these rules to find the derivative of f(x) = xcos(x) + sin(x). We'll break down the process into manageable steps:

1. Apply the Sum Rule: Since f(x) is a sum of two terms, xcos(x) and sin(x), we can find the derivative of each term separately and add the results:

   f'(x) = d/dx (xcos(x)) + d/dx (sin(x))

2. Differentiate xcos(x) using the Product Rule: The term xcos(x) is a product of two functions, x and cos(x). Applying the product rule:

   d/dx (xcos(x)) = (d/dx (x))cos(x) + x(d/dx (cos(x)))

   d/dx (x) = 1 and d/dx (cos(x)) = -sin(x). Substituting these:

   d/dx (xcos(x)) = 1cos(x) + x(-sin(x)) = cos(x) - xsin(x)

3. Differentiate sin(x): The derivative of sin(x) is simply cos(x):

   d/dx (sin(x)) = cos(x)

4. Combine the Results: Substituting the derivatives from steps 2 and 3 back into step 1:

   f'(x) = (cos(x) - xsin(x)) + cos(x)

5. Simplify: Combining like terms, we get the final derivative:

   f'(x) = 2cos(x) - xsin(x)

Therefore, the derivative of xcos(x) + sin(x) is 2cos(x) - xsin(x).

Detailed Explanation and Justification

The steps above demonstrate the application of fundamental calculus rules. Let's delve into a more detailed justification for each step:

• Step 1 (Sum Rule): The linearity of differentiation allows us to treat the derivative of a sum as the sum of the derivatives. This is a direct consequence of the limit properties.

• Step 2 (Product Rule): The product rule arises from the need to account for the change in both functions when considering the derivative of their product. It's not simply the product of their individual derivatives; it incorporates the contribution of each function's change multiplied by the other function. Intuitively, imagine the area of a rectangle; if both length and width are changing, the change in area depends on both the change in length and width, weighted by the other dimension.

• Step 3 (Derivative of sin(x)): This is a standard result derived from the definition of the derivative and the limit of trigonometric functions.

• Step 4 & 5 (Combination and Simplification): These steps are straightforward algebraic manipulation to present the final derivative in a concise form.

Higher-Order Derivatives

The process of differentiation can be repeated to find higher-order derivatives. For instance, the second derivative, f''(x), is the derivative of f'(x). Let's find the second derivative of our function:

f'(x) = 2cos(x) - xsin(x)

Applying the sum and product rules again:

f''(x) = d/dx (2cos(x)) - d/dx (xsin(x))

f''(x) = -2sin(x) - (sin(x) + xcos(x))

f''(x) = -3sin(x) - xcos(x)

Thus, the second derivative is -3sin(x) - xcos(x). This process can be continued to find third, fourth, and higher-order derivatives.

Applications and Significance

The derivative of xcos(x) + sin(x), and its higher-order derivatives, have various applications in mathematics, physics, and engineering. Some examples include:

• Optimization Problems: Finding critical points (maxima and minima) of a function requires setting its first derivative equal to zero and solving for x.

• Curve Sketching: The first and second derivatives provide information about the function's increasing/decreasing intervals and concavity, aiding in accurate curve sketching.

• Physics: In physics, derivatives often represent rates of change. For example, the velocity is the derivative of displacement with respect to time, and acceleration is the derivative of velocity. Functions similar to xcos(x) + sin(x) might model oscillatory phenomena.

• Signal Processing: In signal processing, derivatives are employed in analyzing and manipulating signals. Trigonometric functions are frequently used to represent periodic signals, and their derivatives are crucial for understanding their frequency components.

Frequently Asked Questions (FAQ)

• Q: Why is the product rule necessary here? A: The product rule is essential because the term xcos(x) involves the product of two functions, x and cos(x). The derivative of a product isn't simply the product of the derivatives.

• Q: Can I use a different method to find the derivative? A: While the method shown is the most straightforward, other techniques might be employed depending on the context, such as logarithmic differentiation for some cases. However, for this function, the sum and product rules are the most efficient and readily applicable.

• Q: What is the significance of the second derivative? A: The second derivative provides information about the concavity of the function (whether it's concave up or concave down). It also signifies the rate of change of the first derivative (which represents the rate of change of the original function).

• Q: Are there any limitations to this method? A: The method is generally applicable to functions of this form. However, for extremely complex functions, more advanced techniques or computational tools might be required.

Conclusion

Finding the derivative of xcos(x) + sin(x) involves applying fundamental rules of differentiation, namely the sum and product rules, along with the derivatives of trigonometric functions. This process not only provides the answer—2cos(x) - xsin(x)—but also illustrates the power and practicality of these calculus principles. Understanding these methods lays a strong foundation for tackling more challenging differentiation problems and comprehending the rich applications of derivatives in various fields. Remember to practice applying these rules to build your understanding and proficiency in calculus. The journey of mastering calculus is iterative and rewarding, so keep exploring and expanding your knowledge.