Find The Indicated Partial Derivative

Finding the Indicated Partial Derivative: A Comprehensive Guide

Finding partial derivatives is a fundamental concept in multivariable calculus. Understanding how to calculate them is crucial for various applications, from optimizing functions in economics and engineering to understanding rates of change in physics and biology. This comprehensive guide will walk you through the process, covering various techniques and providing examples to solidify your understanding. We'll explore different notations, delve into the underlying principles, and tackle more complex scenarios, equipping you with the skills to confidently find any indicated partial derivative.

Understanding Partial Derivatives

Before diving into the mechanics, let's clarify the concept. A partial derivative measures the rate of change of a multivariable function with respect to one of its variables, while holding all other variables constant. Imagine a mountain's elevation (our function) changing as you move east and north. A partial derivative would tell you how fast the elevation changes if you only moved east, keeping your north-south position fixed.

Consider a function of two variables, f(x, y). The partial derivative with respect to x, denoted as ∂f/∂x, ∂_xf, or f_x, represents the instantaneous rate of change of f as x changes, with y held constant. Similarly, ∂f/∂y, ∂_yf, or f_y represents the instantaneous rate of change of f as y changes, holding x constant. This extends to functions with three or more variables.

Calculating Partial Derivatives: A Step-by-Step Approach

Calculating partial derivatives is surprisingly straightforward. You treat the variable you're not differentiating with respect to as a constant. Let's illustrate with examples:

Example 1: A Simple Polynomial

Let f(x, y) = 3x² + 2xy - y³. Find ∂f/∂x and ∂f/∂y.

• Finding ∂f/∂x: Treat y as a constant. The derivative of 3x² with respect to x is 6x. The derivative of 2xy (treating y as a constant) is 2y. The derivative of -y³ with respect to x is 0 (since -y³ is a constant). Therefore, ∂f/∂x = 6x + 2y.

• Finding ∂f/∂y: Treat x as a constant. The derivative of 3x² with respect to y is 0. The derivative of 2xy (treating x as a constant) is 2x. The derivative of -y³ with respect to y is -3y². Therefore, ∂f/∂y = 2x - 3y².

Example 2: Incorporating Exponential and Trigonometric Functions

Let f(x, y) = e^(xy)sin(x). Find ∂f/∂x and ∂f/∂y.

• Finding ∂f/∂x: This requires the product rule and the chain rule. Remember, y is treated as a constant.

  ∂f/∂x = [ye^(xy)]sin(x) + e^(xy)cos(x)

• Finding ∂f/∂y: Here, x is treated as a constant.

  ∂f/∂y = xe^(xy)sin(x)

Example 3: A Function with Three Variables

Let f(x, y, z) = x²yz + ln(x + y²). Find ∂f/∂x, ∂f/∂y, and ∂f/∂z.

• Finding ∂f/∂x: Treat y and z as constants.

  ∂f/∂x = 2xyz + 1/(x + y²)

• Finding ∂f/∂y: Treat x and z as constants.

  ∂f/∂y = x²z + 2y/(x + y²)

• Finding ∂f/∂z: Treat x and y as constants.

  ∂f/∂z = x²y

Higher-Order Partial Derivatives

Just as with single-variable calculus, you can find second-order, third-order, and even higher-order partial derivatives. These involve taking partial derivatives of partial derivatives.

For instance, for a function f(x, y):

• Second-order partial derivatives:
  • ∂²f/∂x² (or f_xx): The partial derivative of ∂f/∂x with respect to x.
  • ∂²f/∂y² (or f_yy): The partial derivative of ∂f/∂y with respect to y.
  • ∂²f/∂x∂y (or f_xy): The partial derivative of ∂f/∂x with respect to y (also called a mixed partial derivative).
  • ∂²f/∂y∂x (or f_yx): The partial derivative of ∂f/∂y with respect to x (another mixed partial derivative).

Under certain conditions (specifically, if the function has continuous second-order partial derivatives), Clairaut's Theorem states that ∂²f/∂x∂y = ∂²f/∂y∂x. This means the order of differentiation doesn't matter.

Example 4: Second-Order Partial Derivatives

Let f(x, y) = x³y² + sin(x). Find ∂²f/∂x², ∂²f/∂y², ∂²f/∂x∂y, and ∂²f/∂y∂x.

• ∂f/∂x = 3x²y² + cos(x)
• ∂f/∂y = 2x³y
• ∂²f/∂x² = 6xy² - sin(x)
• ∂²f/∂y² = 2x³
• ∂²f/∂x∂y = 6x²y
• ∂²f/∂y∂x = 6x²y (Note: ∂²f/∂x∂y = ∂²f/∂y∂x as expected by Clairaut's Theorem)

Implicit Differentiation and Partial Derivatives

When dealing with implicitly defined functions, where the variables are not explicitly expressed in terms of each other (e.g., x² + y² = r²), we use implicit differentiation. The process remains similar for partial derivatives: differentiate with respect to one variable, treating the others as constants, and then solve for the desired partial derivative.

Example 5: Implicit Differentiation

Given x² + y² + z² = 1, find ∂z/∂x and ∂z/∂y.

• To find ∂z/∂x, differentiate the equation with respect to x, treating y as a constant:

  2x + 2z(∂z/∂x) = 0 => ∂z/∂x = -x/z

• To find ∂z/∂y, differentiate with respect to y, treating x as a constant:

  2y + 2z(∂z/∂y) = 0 => ∂z/∂y = -y/z

Applications of Partial Derivatives

The applications of partial derivatives are vast and permeate many fields:

• Optimization: Finding maximum and minimum values of multivariable functions (e.g., maximizing profit in economics, minimizing material usage in engineering).
• Gradient Vector: The gradient vector, composed of partial derivatives, points in the direction of the steepest ascent of a function.
• Directional Derivatives: Measure the rate of change of a function in a specific direction.
• Physics and Engineering: Modeling changes in temperature, pressure, velocity, and other physical quantities.
• Economics: Analyzing marginal productivity, utility functions, and market equilibrium.

Frequently Asked Questions (FAQ)

Q1: What is the difference between a partial derivative and a total derivative?

A total derivative considers the change in a function with respect to all its variables simultaneously, while a partial derivative only considers the change with respect to one variable while holding others constant.

Q2: Can I use the quotient rule for partial derivatives?

Yes, the quotient rule applies to partial derivatives just as it does to single-variable derivatives. Remember to treat the variables not being differentiated with respect to as constants.

Q3: What if a function is not differentiable at a certain point?

If a function is not differentiable at a particular point, the partial derivatives may not exist at that point. This can occur at points where the function is discontinuous, has sharp corners, or has other irregularities.

Conclusion

Finding indicated partial derivatives is a cornerstone of multivariable calculus. Through understanding the fundamental concept of treating other variables as constants and mastering the various differentiation techniques, you can successfully tackle a wide range of problems. From simple polynomials to complex implicit functions, the systematic approach outlined in this guide will equip you with the tools to confidently calculate partial derivatives of any order. Remember to practice consistently, and you'll quickly develop proficiency in this essential mathematical skill. The applications are diverse and rewarding, opening doors to advanced studies and problem-solving in numerous fields.