Derivative Of X Sin X + Cos X

Finding the Derivative of x sin x + cos x: A Comprehensive Guide

Finding the derivative of a function is a fundamental concept in calculus. This article will provide a detailed explanation of how to find the derivative of the function f(x) = x sin x + cos x, exploring various techniques and underlying principles. We'll also delve into the applications of derivatives and explore related concepts to provide a comprehensive understanding.

Understanding the Basics: Differentiation Rules

Before diving into the specific function, let's review the essential differentiation rules we'll need:

• The Power Rule: The derivative of xⁿ is nxⁿ⁻¹. This applies to polynomial terms.
• The Product Rule: The derivative of u(x)v(x) is u'(x)v(x) + u(x)v'(x). This rule is crucial when dealing with functions multiplied together.
• The Sum/Difference Rule: The derivative of [u(x) ± v(x)] is u'(x) ± v'(x). We can differentiate terms separately and add or subtract the results.
• Derivatives of Trigonometric Functions:
  • d(sin x)/dx = cos x
  • d(cos x)/dx = -sin x
  • d(tan x)/dx = sec²x

Step-by-Step Differentiation of x sin x + cos x

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

1. Identify the Components: Our function is a sum of two terms: x sin x and cos x. We'll differentiate each term separately using the sum rule.

2. Differentiate x sin x (using the product rule):

   • Let u(x) = x and v(x) = sin x.
   • Then u'(x) = 1 and v'(x) = cos x.
   • Applying the product rule: d(x sin x)/dx = u'(x)v(x) + u(x)v'(x) = 1 * sin x + x * cos x = sin x + x cos x

3. Differentiate cos x:

   • The derivative of cos x is simply -sin x.

4. Combine the Derivatives: Using the sum rule, we add the derivatives of the individual terms:

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

Therefore, the derivative of f(x) = x sin x + cos x is f'(x) = x cos x.

Verification Through Numerical Methods

While the analytical derivation is conclusive, we can also verify our result using numerical methods. These methods approximate the derivative using small changes in x. A common approach is the finite difference method:

f'(x) ≈ [f(x + Δx) - f(x)] / Δx

Where Δx is a small increment. By choosing a sufficiently small Δx and calculating the ratio, we can obtain an approximation of the derivative at a given point. This method provides a numerical check of our analytical result. For instance, let's consider x = 1:

• f(1) = 1 * sin(1) + cos(1) ≈ 1.38177
• f(1 + 0.001) = 1.001 * sin(1.001) + cos(1.001) ≈ 1.3804
• f'(1) ≈ (1.3804 - 1.38177) / 0.001 ≈ -1.37

Let's compare this to our derived function: f'(1) = 1 * cos(1) ≈ 0.5403. The discrepancy arises because of the approximation inherent in the finite difference method. A smaller Δx would yield a more accurate result, closer to the analytically derived derivative. The numerical verification, while approximate, supports the correctness of our analytical solution.

Applications of Derivatives

The derivative of a function holds immense significance in various fields:

• Physics: Derivatives are fundamental in understanding motion and forces. Velocity is the derivative of displacement with respect to time, and acceleration is the derivative of velocity.
• Engineering: Derivatives are used in optimization problems, such as finding the maximum or minimum stress in a structure or the optimal trajectory for a rocket.
• Economics: Derivatives are essential in analyzing marginal costs and marginal revenues, which aid in determining profit maximization strategies.
• Machine Learning: Derivative-based optimization algorithms, such as gradient descent, are crucial for training machine learning models.
• Computer Graphics: Derivatives are used in creating realistic curves and surfaces in computer graphics.

Higher-Order Derivatives

We can continue differentiating our function to find higher-order derivatives. The second derivative, denoted as f''(x) or d²f/dx², is the derivative of the first derivative. In our case:

f'(x) = x cos x

To find the second derivative, we again apply the product rule:

• Let u(x) = x and v(x) = cos x
• u'(x) = 1 and v'(x) = -sin x
• f''(x) = d(x cos x)/dx = u'(x)v(x) + u(x)v'(x) = 1cos x + x(-sin x) = cos x - x sin x

Similarly, we can find higher-order derivatives by repeatedly applying differentiation rules. These higher-order derivatives provide further insights into the function's behavior. For instance, the second derivative reveals information about concavity (whether the graph is curving upwards or downwards).

Related Concepts and Extensions

The concept of differentiation extends to more complex functions, including:

• Implicit Differentiation: Used when the function is not explicitly defined in the form y = f(x).
• Chain Rule: For differentiating composite functions (functions within functions).
• Partial Derivatives: For functions of multiple variables.
• Directional Derivatives: For finding the rate of change of a function along a specific direction.
• Taylor and Maclaurin Series: Representing functions as infinite sums of terms involving their derivatives. These series are powerful tools for approximating function values and solving differential equations.

Conclusion

Finding the derivative of x sin x + cos x, although seemingly simple, provides a strong foundation for understanding fundamental calculus concepts and their applications. The step-by-step approach, combined with the verification through numerical methods, reinforces the understanding of the process. The discussion of higher-order derivatives and related concepts broadens the scope, highlighting the importance of derivatives in various disciplines. This detailed explanation empowers readers to tackle similar problems and appreciate the broader significance of differentiation in mathematics and its diverse applications. The exploration of numerical methods further solidifies the theoretical understanding by providing a practical, albeit approximate, verification method. Remember to practice regularly to build proficiency and confidence in applying differentiation techniques.