Problem 42 - Calculus Early Transcendentals: Pearson New International Edition

Question

If \(f(x, y, z)=(x y / z)^{1 / 2}\), find \(f_{x}(-2,-1,8)\).

Step-by-Step Solution

Answer

The partial derivative \( f_x(-2, -1, 8) \) is \( \frac{-1}{8} \).

Step 1: Understand the Exercise

We are given the function \( f(x, y, z) = \left( \frac{x y}{z} \right)^{1/2} \) and asked to find the partial derivative of \( f \) with respect to \( x \) at the point \((-2, -1, 8)\).

Step 2: Setup Partial Derivative

The partial derivative of \( f(x, y, z) \) with respect to \( x \) is given by \( f_x(x, y, z) = \frac{\partial}{\partial x}\left( \frac{x y}{z} \right)^{1/2} \).

Step 3: Apply Chain Rule

To differentiate \( \left( \frac{x y}{z} \right)^{1/2} \) with respect to \( x \), we'll use the chain rule. Let \( u = \frac{x y}{z} \), thus \( f(x, y, z) = u^{1/2} \). The derivative \( \frac{d}{dx} u^{1/2} = \frac{1}{2}u^{-1/2} \cdot \frac{du}{dx} \).

Step 4: Differentiate the Inner Function

The derivative of \( u = \frac{x y}{z} \) with respect to \( x \) is \( \frac{y}{z} \), since \( y \) and \( z \) are treated as constants in partial differentiation with respect to \( x \).

Step 5: Calculate Derivative at the Point

Substitute \( x = -2 \), \( y = -1 \), and \( z = 8 \) into the formula. First, calculate \( u = \frac{x y}{z} = \frac{-2 \cdot (-1)}{8} = \frac{1}{4} \). Then substitute into the derivative, \( f_x(x, y, z) = \frac{1}{2}(\frac{1}{4})^{-1/2} \cdot \frac{-1}{8} = \frac{1}{2} \cdot 2 \cdot \frac{-1}{8} = \frac{-1}{8} \).

Step 6: Final Answer

The partial derivative \( f_x(-2, -1, 8) \) is \( \frac{-1}{8} \).

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept in multivariable calculus, particularly for functions of several variables like \( f(x, y, z) = (\frac{xy}{z})^{1/2} \). In essence, a partial derivative measures the rate at which a function changes as one of its variables is changed, while keeping other variables constant.

Taking the partial derivative with respect to \( x \) of our given function means evaluating how the function changes as \( x \) changes, treating \( y \) and \( z \) as constants. We find \( f_x(x, y, z) \), which is expressed as \( \frac{\partial}{\partial x}(\frac{xy}{z})^{1/2} \).

The process involves the following steps:

• First, identify the expression inside the square root: \( \frac{xy}{z} \).
• Apply the chain rule to differentiate with respect to \( x \), as detailed next.

Ultimately, this helps us find the specific rate of change of the function at any given point, such as (-2, -1, 8). Understanding partial derivatives enhances our ability to analyze how individual variables affect multivariable functions.

Chain Rule

In the context of partial differentiation, the chain rule is crucial when dealing with composite functions. Our function \((\frac{xy}{z})^{1/2}\) can be seen as a composition of the inner function \( u = \frac{xy}{z} \) and the outer function \( u^{1/2} \).

To differentiate this composite function concerning \( x \), we apply the chain rule:

• Differentiate the outer function: For \( u^{1/2} \), this gives \( \frac{1}{2}u^{-1/2} \).
• Then, differentiate the inner function \( u = \frac{xy}{z} \) with respect to \( x \). Since \( y \) and \( z \) are constants, the derivative is \( \frac{y}{z} \).

Thus, applying the chain rule results in:\[\frac{d}{dx} (\frac{xy}{z})^{1/2} = \frac{1}{2}(\frac{xy}{z})^{-1/2} \cdot \frac{y}{z}.\]The chain rule provides a systematic method to handle differentiation for layered functions, ensuring accurate and efficient results.

Functions of Several Variables

Functions of several variables, such as \( f(x, y, z) = (\frac{xy}{z})^{1/2} \), are integral to modeling real-world scenarios involving multiple influencing factors. Each variable represents a dimension, allowing the function to describe a surface or shape in a multidimensional space.

In practical applications:

• Variables often correspond to different physical quantities (e.g., time, distance, temperature).
• The function's value reflects a property influenced by these quantities, providing insights into how changes in each variable affect the overall outcome.

Understanding these functions requires familiarity with concepts like partial derivatives, as each variable's change impacts the function distinctly. Navigating these complexities requires tools like the chain rule and other differentiation techniques, ensuring we can describe and predict influences in dynamic systems.