Find All The Second Order Partial Derivatives Of

Here's a comprehensive article that dives into finding all the second-order partial derivatives of a function. This article aims to provide a clear, step-by-step guide with examples, explanations, and practical insights.

Mastering Second-Order Partial Derivatives: A Comprehensive Guide

Partial derivatives are fundamental in multivariable calculus, offering insights into how a function changes with respect to one variable while holding others constant. Expanding on this, second-order partial derivatives delve deeper, revealing information about the rate of change of the rate of change – essentially, the concavity or curvature of the function in different directions. Understanding and calculating these derivatives is crucial for various applications, from optimization problems to analyzing physical phenomena.

This article will guide you through the process of finding all the second-order partial derivatives of a function, providing clear explanations, step-by-step instructions, and practical examples.

Introduction

Imagine a landscape where the height varies depending on your position (both latitude and longitude). Partial derivatives tell you how much the height changes as you move slightly east or north. Second-order partial derivatives tell you how that rate of change itself is changing – is the slope becoming steeper or flatter as you move in those directions?

The concept of second-order partial derivatives builds on the foundation of first-order partial derivatives. Before diving into second-order derivatives, let's quickly recap first-order partial derivatives. For a function f(x, y), the first-order partial derivatives are:

• ∂f/∂x: The partial derivative of f with respect to x, treating y as a constant.
• ∂f/∂y: The partial derivative of f with respect to y, treating x as a constant.

These first-order derivatives tell us the instantaneous rate of change of f in the x and y directions, respectively. Now, let's extend this idea to second-order derivatives.

Defining Second-Order Partial Derivatives

Second-order partial derivatives are simply the partial derivatives of the first-order partial derivatives. For a function f(x, y), there are four possible second-order partial derivatives:

• ∂²f/∂x²: The partial derivative of ∂f/∂x with respect to x. This is also written as f_xx. It represents how the rate of change of f with respect to x changes as x varies.

• ∂²f/∂y²: The partial derivative of ∂f/∂y with respect to y. This is also written as f_yy. It represents how the rate of change of f with respect to y changes as y varies.

• ∂²f/∂x∂y: The partial derivative of ∂f/∂y with respect to x. This is also written as f_yx. This is a mixed partial derivative, showing how the rate of change of f with respect to y changes as x varies.

• ∂²f/∂y∂x: The partial derivative of ∂f/∂x with respect to y. This is also written as f_xy. This is also a mixed partial derivative, showing how the rate of change of f with respect to x changes as y varies.

Clairaut's Theorem: A crucial theorem to remember is Clairaut's Theorem (also known as Schwarz's Theorem). It states that if the second-order mixed partial derivatives are continuous at a point, then they are equal:

∂²f/∂x∂y = ∂²f/∂y∂x   (or  f<sub>yx</sub> = f<sub>xy</sub>)

This theorem significantly simplifies calculations, as you only need to compute one of the mixed partial derivatives if you know the function satisfies the continuity condition. In most practical cases, especially with well-behaved functions, this condition holds true.

Step-by-Step Guide to Finding Second-Order Partial Derivatives

Let's break down the process into a clear, step-by-step method:

1. Find the First-Order Partial Derivatives:

• Calculate ∂f/∂x by differentiating f(x, y) with respect to x, treating y as a constant.
• Calculate ∂f/∂y by differentiating f(x, y) with respect to y, treating x as a constant.

2. Find the Second-Order Partial Derivatives:

• Calculate ∂²f/∂x² by differentiating ∂f/∂x with respect to x.
• Calculate ∂²f/∂y² by differentiating ∂f/∂y with respect to y.
• Calculate ∂²f/∂x∂y by differentiating ∂f/∂y with respect to x.
• Calculate ∂²f/∂y∂x by differentiating ∂f/∂x with respect to y.

3. Verify Clairaut's Theorem (Optional but Recommended):

• If possible (and if not too computationally expensive), compare ∂²f/∂x∂y and ∂²f/∂y∂x. If they are equal (or very close, allowing for minor calculation errors), it provides confidence in your calculations.

Examples with Detailed Solutions

Let's illustrate this process with several examples:

Example 1: A Simple Polynomial Function

Let f(x, y) = x³ + 2xy² + y³

1. First-Order Partial Derivatives:

   • ∂f/∂x = 3x² + 2y²
   • ∂f/∂y = 4xy + 3y²

2. Second-Order Partial Derivatives:

   • ∂²f/∂x² = ∂/∂x (3x² + 2y²) = 6x
   • ∂²f/∂y² = ∂/∂y (4xy + 3y²) = 4x + 6y
   • ∂²f/∂x∂y = ∂/∂x (4xy + 3y²) = 4y
   • ∂²f/∂y∂x = ∂/∂y (3x² + 2y²) = 4y

3. Verification of Clairaut's Theorem:

   • ∂²f/∂x∂y = 4y and ∂²f/∂y∂x = 4y. They are equal, confirming Clairaut's Theorem.

Example 2: An Exponential Function

Let f(x, y) = e^{(x² + y)}

1. First-Order Partial Derivatives:

   • ∂f/∂x = 2xe^{(x² + y)}
   • ∂f/∂y = e^{(x² + y)}

2. Second-Order Partial Derivatives:

   • ∂²f/∂x² = ∂/∂x (2xe^{(x² + y)}) = 2e^{(x² + y)} + 4x²e^{(x² + y)} = (2 + 4x²)e^{(x² + y)}
   • ∂²f/∂y² = ∂/∂y (e^{(x² + y)}) = e^{(x² + y)}
   • ∂²f/∂x∂y = ∂/∂x (e^{(x² + y)}) = 2xe^{(x² + y)}
   • ∂²f/∂y∂x = ∂/∂y (2xe^{(x² + y)}) = 2xe^{(x² + y)}

3. Verification of Clairaut's Theorem:

   • ∂²f/∂x∂y = 2xe^{(x² + y)} and ∂²f/∂y∂x = 2xe^{(x² + y)}. They are equal, confirming Clairaut's Theorem.

Example 3: A Trigonometric Function

Let f(x, y) = sin(x)cos(y)

1. First-Order Partial Derivatives:

   • ∂f/∂x = cos(x)cos(y)
   • ∂f/∂y = -sin(x)sin(y)

2. Second-Order Partial Derivatives:

   • ∂²f/∂x² = ∂/∂x (cos(x)cos(y)) = -sin(x)cos(y)
   • ∂²f/∂y² = ∂/∂y (-sin(x)sin(y)) = -sin(x)cos(y)
   • ∂²f/∂x∂y = ∂/∂x (-sin(x)sin(y)) = -cos(x)sin(y)
   • ∂²f/∂y∂x = ∂/∂y (cos(x)cos(y)) = -cos(x)sin(y)

3. Verification of Clairaut's Theorem:

   • ∂²f/∂x∂y = -cos(x)sin(y) and ∂²f/∂y∂x = -cos(x)sin(y). They are equal, confirming Clairaut's Theorem.

Example 4: A More Complex Function (Quotient Rule Required)

Let f(x, y) = x / (x² + y²)

1. First-Order Partial Derivatives:

   • ∂f/∂x = [(x² + y²)(1) - x(2x)] / (x² + y²)² = (y² - x²) / (x² + y²)²
   • ∂f/∂y = [0 - x(2y)] / (x² + y²)² = -2xy / (x² + y²)²

2. Second-Order Partial Derivatives:

   • ∂²f/∂x² = ∂/∂x [(y² - x²) / (x² + y²)²] = [(-2x)(x² + y²)² - (y² - x²)(2)(x² + y²)(2x)] / (x² + y²)⁴ = [(-2x)(x² + y²) - 4x(y² - x²)] / (x² + y²)³ = [(-2x³ - 2xy²) - (4xy² - 4x³)] / (x² + y²)³ = (2x³ - 6xy²) / (x² + y²)³
   • ∂²f/∂y² = ∂/∂y [-2xy / (x² + y²)²] = [(-2x)(x² + y²)² - (-2xy)(2)(x² + y²)(2y)] / (x² + y²)⁴ = [(-2x)(x² + y²) + 8xy²] / (x² + y²)³ = (-2x³ + 6xy²) / (x² + y²)³
   • ∂²f/∂x∂y = ∂/∂x [-2xy / (x² + y²)²] = [(-2y)(x² + y²)² - (-2xy)(2)(x² + y²)(2x)] / (x² + y²)⁴ = [(-2y)(x² + y²) + 8x²y] / (x² + y²)³ = (6x²y - 2y³) / (x² + y²)³
   • ∂²f/∂y∂x = ∂/∂y [(y² - x²) / (x² + y²)²] = [(2y)(x² + y²)² - (y² - x²)(2)(x² + y²)(2y)] / (x² + y²)⁴ = [(2y)(x² + y²) - 4y(y² - x²)] / (x² + y²)³ = (6x²y - 2y³) / (x² + y²)³

3. Verification of Clairaut's Theorem:

   • ∂²f/∂x∂y = (6x²y - 2y³) / (x² + y²)³ and ∂²f/∂y∂x = (6x²y - 2y³) / (x² + y²)³. They are equal, confirming Clairaut's Theorem. Note that this example is more complex and prone to errors. Careful application of the quotient rule is essential.

Practical Applications

Second-order partial derivatives are crucial in several areas:

• Optimization: In optimization problems, the second-order partial derivatives help determine whether a critical point is a local maximum, local minimum, or saddle point. The Hessian matrix, which contains the second-order partial derivatives, is used to perform the second derivative test for multivariable functions.

• Physics and Engineering: They appear in wave equations, heat equations, and fluid dynamics, describing how quantities like temperature, displacement, or pressure change over space and time.

• Economics: In economics, they are used to analyze the concavity and convexity of utility functions, production functions, and cost functions. This helps economists understand concepts like diminishing returns and economies of scale.

• Machine Learning: Second-order derivatives are used in optimization algorithms, particularly in training neural networks. Hessian-free optimization methods leverage second-order information to improve convergence speed.

Common Mistakes to Avoid

• Forgetting the Chain Rule: When differentiating composite functions, always remember to apply the chain rule correctly. This is especially important when dealing with exponential, logarithmic, or trigonometric functions.

• Treating Variables Incorrectly: Ensure you are treating the correct variables as constants when taking partial derivatives. Double-check your work to avoid accidentally differentiating with respect to the wrong variable.

• Algebraic Errors: Complex functions often involve lengthy algebraic manipulations. Be meticulous in your calculations to avoid errors in simplification and expansion.

• Ignoring Clairaut's Theorem: While not strictly an error, failing to verify Clairaut's Theorem (when applicable) can be a missed opportunity to catch mistakes.

Tips for Success

• Practice Regularly: The more you practice, the more comfortable you'll become with the process. Work through a variety of examples with varying levels of complexity.

• Break Down Complex Problems: If a function is particularly complicated, break the problem down into smaller, more manageable steps. Calculate the first-order derivatives separately before moving on to the second-order derivatives.

• Double-Check Your Work: Carefully review each step of your calculations to minimize the risk of errors. Consider using a computer algebra system (CAS) like Mathematica or SymPy to verify your results.

• Understand the Concepts: Don't just memorize the formulas. Take the time to understand the underlying concepts and principles. This will help you apply the techniques correctly and solve problems more effectively.

FAQ (Frequently Asked Questions)

• Q: What is the difference between ∂²f/∂x∂y and ∂²f/∂y∂x?

  • A: ∂²f/∂x∂y is the partial derivative of ∂f/∂y with respect to x, while ∂²f/∂y∂x is the partial derivative of ∂f/∂x with respect to y. Clairaut's Theorem states that if these mixed partial derivatives are continuous, they are equal.

• Q: When is Clairaut's Theorem applicable?

  • A: Clairaut's Theorem is applicable when the second-order mixed partial derivatives are continuous at the point of interest. In practice, this condition is usually met for most well-behaved functions.

• Q: How do I find the second-order partial derivatives of a function with three variables, like f(x, y, z)?

  • A: The process is similar. You would first find the first-order partial derivatives (∂f/∂x, ∂f/∂y, ∂f/∂z), and then differentiate each of those with respect to x, y, and z again. This would result in nine second-order partial derivatives (including mixed derivatives).

• Q: What is the Hessian matrix, and how is it related to second-order partial derivatives?

  • A: The Hessian matrix is a square matrix of second-order partial derivatives of a function. For a function f(x, y), the Hessian matrix is:

  H = | ∂²f/∂x²   ∂²f/∂x∂y |
      | ∂²f/∂y∂x   ∂²f/∂y² |
  

  The Hessian matrix is used in the second derivative test to determine the nature of critical points (local maximum, local minimum, or saddle point).

Conclusion

Finding second-order partial derivatives is a fundamental skill in multivariable calculus with broad applications across various fields. By following the step-by-step guide, practicing with examples, and understanding the underlying concepts, you can master this technique and apply it effectively to solve complex problems. Remember to leverage Clairaut's Theorem when applicable and double-check your work to avoid common errors. With consistent effort and a solid understanding of the principles, you'll be well-equipped to tackle even the most challenging second-order partial derivative problems.

How do you plan to apply your understanding of second-order partial derivatives in your field of study or work? What strategies do you find most helpful when calculating these derivatives?