Problem 15 - Calculus Early Transcendentals

Question

Find the first partial derivatives of the function. $$f(x, y)=y^{5}-3 x y$$

Step-by-Step Solution

Answer

\( \frac{\partial f}{\partial x} = -3y \) and \( \frac{\partial f}{\partial y} = 5y^4 - 3x \).

Step 1: Identify the Variables

The function given is \( f(x, y) = y^5 - 3xy \). It is a function of two variables, \(x\) and \(y\). To find the first partial derivatives, we need to treat each variable separately while treating the other variable as a constant.

Step 2: Find the Partial Derivative with Respect to x

To find the partial derivative of \( f(x, y) \) with respect to \( x \), treat \( y \) as a constant. Differentiate each term accordingly: 1. The derivative of \( y^5 \) with respect to \( x \) is 0, since \( y^5 \) is a constant with respect to \( x \).2. The derivative of \(-3xy\) with respect to \( x \) is \(-3y\) (since the derivative of \( x \) with respect to \( x \) is 1).Thus, \( \frac{\partial f}{\partial x} = -3y \).

Step 3: Find the Partial Derivative with Respect to y

To find the partial derivative of \( f(x, y) \) with respect to \( y \), treat \( x \) as a constant. Differentiate each term accordingly:1. The derivative of \( y^5 \) with respect to \( y \) is \( 5y^4 \), using the power rule.2. The derivative of \(-3xy\) with respect to \( y \) is \(-3x\) (since \( x \) is treated as a constant and the derivative of \( y \) is 1).Thus, \( \frac{\partial f}{\partial y} = 5y^4 - 3x \).

Key Concepts

Multivariable Calculus

Multivariable calculus deals with functions that depend on more than one variable. In this exercise, the function is denoted as \( f(x, y) = y^5 - 3xy \), a function of the two variables \(x\) and \(y\). This branch of calculus extends concepts like differentiation from single-variable calculus to multiple dimensions. When handling functions with multiple inputs, understanding how changes in each variable can influence the outcome is crucial.

In multivariable calculus, we commonly encounter:

• Partial Derivatives: These indicate how the function changes as each variable is varied individually while keeping the others constant.
• Gradients and Directional Derivatives: These concepts allow us to understand the rate of change in any direction within the domain of the function.

By mastering these tools, you can better analyze and comprehend the behavior of multivariable functions in contexts such as physics, engineering, and economics.

Differentiation

Differentiation is the process of finding the rate at which something changes. In single-variable calculus, it involves taking derivatives. However, in the context of multivariable functions, we've got partial derivatives to consider.

For the function \( f(x, y) \), we compute the partial derivatives by focusing on one variable at a time. Each time we differentiate:

• The variable of focus is treated normally, allowing us to employ standard differentiation rules like the power rule.
• All other variables are considered constants, which simplifies the differentiation of their terms to zero.

This approach helps simplify our calculations and adds clarity when determining how each variable affects the overall function. For example, when finding \( \frac{\partial f}{\partial x} \) from \( y^5 - 3xy \), treat \(y\) as a constant and note the changes in function as \(x\) varies.

Chain Rule

The chain rule is a powerful tool in calculus, particularly when dealing with composite functions or functions of functions. In the realm of partial derivatives, the chain rule becomes crucial when functions are dependent on more than one variable, and each of these variables is itself influenced by another variable.

Although the original exercise doesn't directly involve using the chain rule, having a grasp of it is essential for more complex scenarios in multivariable calculus. The chain rule effectively allows us to differentiate a function based on more than one variable, such as:

• Outer and inner functions in composition.
• Functions that depend on parameterized curves or surfaces.

By applying the chain rule accurately, you can dissect intricate dependencies between variables, leading to a deeper understanding of systems modeled through multivariable functions. The rule provides a gateway to unraveling the interplay between interconnected components within a mathematical function.