Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-16, find all first partial derivatives of each function. \(f(x, y)=\left(4 x-y^{2}\right)^{3 / 2}\)

Step-by-Step Solution

Answer

The partial derivatives are: \( f_x(x, y) = 6(4x - y^2)^{1/2} \) and \( f_y(x, y) = -3y(4x - y^2)^{1/2} \).

Step 1: Identify Variables

The function is given as \( f(x, y) = (4x - y^2)^{3/2} \). Here, the variables are \( x \) and \( y \). We are asked to find the partial derivatives with respect to each of these variables.

Step 2: Differentiate with Respect to x

To find the partial derivative of \( f \) with respect to \( x \), denote it as \( f_x(x, y) \). We use the chain rule for differentiation:\[ f_x(x, y) = \frac{d}{dx} \left( (4x - y^2)^{3/2} \right) = \frac{3}{2}(4x - y^2)^{1/2} \cdot \frac{d}{dx}(4x - y^2).\]Since \( \frac{d}{dx}(4x - y^2) = 4 \), we have:\[f_x(x, y) = \frac{3}{2}(4x - y^2)^{1/2} \cdot 4 = 6(4x - y^2)^{1/2}.\]

Step 3: Differentiate with Respect to y

To find the partial derivative of \( f \) with respect to \( y \), denote it as \( f_y(x, y) \). Again, use the chain rule:\[f_y(x, y) = \frac{d}{dy} \left( (4x - y^2)^{3/2} \right) = \frac{3}{2}(4x - y^2)^{1/2} \cdot \frac{d}{dy}(-y^2).\]Since \( \frac{d}{dy}(-y^2) = -2y \), we have:\[f_y(x, y) = \frac{3}{2}(4x - y^2)^{1/2} \cdot (-2y) = -3y(4x - y^2)^{1/2}.\]

Step 4: Conclude the Derivatives

The first partial derivatives of the function are:- With respect to \( x \): \( f_x(x, y) = 6(4x - y^2)^{1/2} \)- With respect to \( y \): \( f_y(x, y) = -3y(4x - y^2)^{1/2} \)

Key Concepts

Chain Rule

The chain rule is a critical concept in calculus, particularly in the differentiation of functions of multiple variables. It helps us find the derivative of a composite function.
In simple terms, the chain rule allows us to see how a change in one variable affects a function, even when that variable itself affects another variable.
For a function like \( f(x, y) = (4x - y^2)^{3/2} \), we view it as a composition of an inner function \( u = 4x - y^2 \) and an outer function \( v = u^{3/2} \). Differentiating such a composite function involves two main steps:

• First, differentiate the outer function \( v \) with respect to \( u \), treating \( u \) as a simple variable.
• Second, multiply that result by the derivative of the inner function \( u \).

This approach breaks down complex differentiation steps into manageable parts, ensuring you accurately find partial derivatives. Using the chain rule effectively simplifies the differentiation process of multi-variable functions.

Differentiation with respect to x

Differentiating with respect to \( x \) means focusing on the variable \( x \) while treating the variable \( y \) as a constant. This is because in partial derivatives, we target one variable at a time to see how changes in it affect the function.
In our function \( f(x, y) = (4x - y^2)^{3/2} \), the partial derivative with respect to \( x \) is denoted as \( f_x(x, y) \).
To compute \( f_x(x, y) \), we:

• Recognize that \( 4x - y^2 \) is treated as the inner function, \( u \), in our composite function.
• Apply the chain rule by taking the derivative of \( u^{3/2} \) with respect to \( x \), which helps us track how changes in \( x \) affect \( u \).

The computation results in \( 6(4x - y^2)^{1/2} \), indicating the rate at which the function \( f \) changes as \( x \) changes, while keeping \( y \) constant.

Differentiation with respect to y

When differentiating with respect to \( y \), we change our focus from \( x \) to \( y \). Here, \( x \) is treated as a constant, assessing the influence \( y \) has on the function independently.
The goal is to find \( f_y(x, y) \), the partial derivative with respect to \( y \). In the function \( f(x, y) = (4x - y^2)^{3/2} \), differentiating with respect to \( y \) involves:

• Again using the chain rule where \( u = 4x - y^2 \) is the inner part.
• Taking the derivative of the outer function \( u^{3/2} \) with respect to \( y \).

The derivative of \( -y^2 \) produces \( -2y \). Multiplying through by the derivative form of the chain rule results in \( -3y(4x - y^2)^{1/2} \).
This value shows how altering \( y \) alone changes \( f \), while ignoring any change in \( x \).

Functions of Multiple Variables

Functions involving more than one variable, such as \( f(x, y) \), are known as functions of multiple variables. They demonstrate how outputs depend on several inputs, highlighting relationships among them.
Partial derivatives are crucial here because they let us explore how each variable individually influences the function.To grasp functions with multiple variables, it’s helpful to understand:

• Each variable independently affects the function, allowing us to isolate effects using partial derivatives.
• The differentiation process reveals how each input variable contributes to changes in function output.

For example, in our function \( f(x, y) = (4x - y^2)^{3/2} \), we examine how \( x \) alone and \( y \) alone influence the function. Each partial derivative provides insights into the specific role and impact of its respective variable.
Such an understanding aids in modeling real-world scenarios where multiple factors influence outcomes, like economics, physics, and engineering contexts.