Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

If \(T=f(x, y, z, w)\) and \(x, y, z\), and \(w\) are each functions of \(s\) and \(t\), write a chain rule for \(\partial T / \partial s\).

Step-by-Step Solution

Answer

\( \frac{\partial T}{\partial s} = \frac{\partial T}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial T}{\partial y} \cdot \frac{\partial y}{\partial s} + \frac{\partial T}{\partial z} \cdot \frac{\partial z}{\partial s} + \frac{\partial T}{\partial w} \cdot \frac{\partial w}{\partial s} \).

Step 1: Identify the Function and Variables

The function is given as \( T = f(x, y, z, w) \), where each variable \( x, y, z, \) and \( w \) is a function of two independent variables \( s \) and \( t \). We need to find the partial derivative of \( T \) with respect to \( s \).

Step 2: Apply the Chain Rule

Since each variable \( x, y, z, \) and \( w \) depends on \( s \), we apply the multi-variable chain rule. The partial derivative of \( T \) with respect to \( s \) is the sum of the partial derivatives of \( T \) with respect to \( x, y, z, \) and \( w \), each multiplied by its respective partial derivative with respect to \( s \).

Step 3: Write the Chain Rule Expression

The expression for the partial derivative \( \frac{\partial T}{\partial s} \) is given by: \[\frac{\partial T}{\partial s} = \frac{\partial T}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial T}{\partial y} \cdot \frac{\partial y}{\partial s} + \frac{\partial T}{\partial z} \cdot \frac{\partial z}{\partial s} + \frac{\partial T}{\partial w} \cdot \frac{\partial w}{\partial s}.\]

Key Concepts

Chain Rule

The **chain rule** is a fundamental concept in calculus. It is especially useful when dealing with functions of multiple variables. The rule enables you to differentiate a function that is composed of other functions. When you have a function like \( T = f(x, y, z, w) \), where each variable is itself a function of other variables, such as \( s \) and \( t \), the chain rule helps compute the derivative of \( T \) with respect to these independent variables.
In the exercise, we are interested in finding the partial derivative of \( T \) with respect to \( s \). This involves breaking down each part. We compute the partial derivatives of \( T \) concerning \( x, y, z, \) and \( w \). Then, we multiply each of these derivatives by the derivative of their respective variable with respect to \( s \). The sum of these products gives us \( \frac{\partial T}{\partial s} \). Effectively, it shows how \( T \) changes as \( s \) changes, considering that \( x, y, z, \) and \( w \) are also influenced by \( s \).
The chain rule in this context is a powerful tool, allowing us to handle complex multi-variable dependencies by breaking them into simpler, manageable parts. By mastering the chain rule, you gain the ability to solve a wide range of problems involving composite functions.

Partial Derivatives

**Partial derivatives** focus on how a function changes as one of its variables changes, holding the other variables constant. When dealing with functions of more than one variable, like \( f(x, y, z, w) \), partial derivatives are extremely helpful.

• They allow one to probe the function's structure by examining how the function varies with respect to a single variable at a time.
• This approach is pivotal in multivariable calculus, as it extends the concept of a derivative to multiple dimensions.

In the original problem, to express \( \frac{\partial T}{\partial s} \), we first calculated the partial derivative of \( T \) with respect to each of its variables \( x, y, z, \) and \( w \). This shows how \( T \) changes with those elements as they vary, without considering other influences.Partial derivatives serve a significant role in applications like analyzing physical models, optimizing systems, and improving complex algorithms. By examining each dimension separately, you can piece together how multiple variables together affect the whole system.

Functions of Multiple Variables

**Functions of multiple variables** involve more than one independent variable. This means that their output depends on two or more inputs. For instance, in the given exercise, \( T \) depends on \( x, y, z, \) and \( w \), each of which is also a function of \( s \) and \( t \).

• The complexity increases when considering how the function behaves regarding multiple independent variables.
• The analysis requires a combination of techniques to understand how the function values are influenced by changes in all its variables.

By dealing with functions of multiple variables, you are looking at how changes in one input affect the output, considering the context of other inputs. These functions form the core of multivariable calculus and are crucial for modeling real-world phenomena like temperature variations across a physical space or changes in financial markets.Understanding functions with multiple variables helps in many fields beyond mathematics, including engineering, physics, and economics. Mastery of these functions provides powerful insights into the behavior of complex systems, offering the ability to predict and explain various phenomena by examining how they respond to different inputs.