Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the following in \(\partial\) notation. (a) \(f_{y y y}\) (b) \(f_{x x y}\) (c) \(f_{x y y y}\)

Step-by-Step Solution

Answer

(a) \(\frac{\partial^3 f}{\partial y^3}\); (b) \(\frac{\partial^3 f}{\partial x^2 \partial y}\); (c) \(\frac{\partial^4 f}{\partial x \partial y^3}\).

Step 1: Understanding the Notation f_{xyz}

The notation like \(f_{yyy}\) or \(f_{xxxy}\) refers to partial derivatives. The subscripts indicate which variable is being derived with respect to, and how many times. \(f_{yyy}\) means the function \(f\) is derived with respect to \(y\) three times.

Step 2: Solving (a): Converting f_{yyy}

The notation \(f_{yyy}\) represents the third partial derivative of \(f\) with respect to \(y\) three times. In \(\partial\) notation, this is written as:\[\frac{\partial^3 f}{\partial y^3}\]

Step 3: Solving (b): Converting f_{xxy}

The notation \(f_{xxy}\) indicates taking the partial derivative of \(f\) first twice with respect to \(x\), and then once with respect to \(y\). In \(\partial\) notation, this is written as:\[\frac{\partial^3 f}{\partial x^2 \partial y}\]

Step 4: Solving (c): Converting f_{xyyy}

The notation \(f_{xyyy}\) represents first taking the partial derivative of \(f\) with respect to \(x\) once, then with respect to \(y\) three times. In \(\partial\) notation, this is written as:\[\frac{\partial^4 f}{\partial x \partial y^3}\]

Key Concepts

Multivariable Calculus

Multivariable Calculus is an extension of single-variable calculus to functions that depend on several variables. This branch of mathematics is crucial for analyzing multi-dimensional systems. For example, when working with 3D space, you might deal with functions of variables like \(x\) and \(y\). Multivariable Calculus allows us to:\

• Investigate how functions change across multi-dimensional fields.
• Find the slopes and curvatures in multiple dimensions.
• Analyze surface areas and volumes of three-dimensional shapes.

In this context, partial derivatives become an essential tool. They help find the rate of change of a function with respect to one variable while keeping other variables constant. This approach is incredibly useful in fields like physics, engineering, and economics, where many variables influence behaviors and outcomes.

Higher-Order Derivatives

Higher-order derivatives involve taking derivatives multiple times. In the realm of multivariable calculus, these derivatives are referred to as partial derivatives. Let's consider a function \(f(x, y)\) depending on two variables. Here are some essentials about higher-order derivatives:\

• First-order Partial Derivative: This is the rate of change of the function with respect to a single variable, like \(\frac{\partial f}{\partial x}\) or \(\frac{\partial f}{\partial y}\).
• Second-order Partial Derivative: This concept involves taking the derivative of a first-order derivative, such as \((\frac{\partial^2 f}{\partial x^2})\), often used to find curvature.
• Third-order or Higher Partial Derivatives: These are derivatives taken to even higher degrees, like \((\frac{\partial^3 f}{\partial y^3})\). These are less common but useful in more complex scenarios.

The procedure for computing higher-order derivatives follows the principle of iteratively applying partial differentiation multiple times as indicated by the subscripts.

Derivative Notation

Derivative Notation provides a way to succinctly represent the operation of differentiation. In single-variable calculus, you might see derivatives written with a simple prime \(f'(x)\) or dy/dx. However, multivariable calculus introduces partial derivative notation. Key points include:\

• Partial Derivative Notation: This uses the \(\partial\) symbol to differentiate functions with respect to one variable. For example, \(\frac{\partial f}{\partial x}\) shows a partial derivative with respect to \(x\).
• Higher-Order Notation: This indicates the number of times a derivative is taken. It's an extension of single-order by incorporating powers, such as \(\frac{\partial^3 f}{\partial y^3}\) for a third-order derivative.
• Combination of Variables: Mixed partial derivatives emerge when functions are differentiated with respect to different variables, like \(\frac{\partial^3 f}{\partial x^2 \partial y}\).

Understanding how to read and write in derivative notation is vital for clarity in expressing the complexity of multivariable functions.