Problem 17 - Calculus Early Transcendentals

Question

\(17-20\) Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. $$u=f(x, y), \quad \text { where } x=x(r, s, t), y=y(r, s, t)$$

Step-by-Step Solution

Answer

The Chain Rule uses partial derivatives to express \( \frac{du}{dr}, \frac{du}{ds}, \frac{du}{dt} \) as linear combinations of \( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \) with \( \frac{\partial x}{\partial}, \frac{\partial y}{\partial} \) of \( r, s, \) and \( t \).

Step 1: Identify the outer function

Given that \( u = f(x, y) \), the outer function here is \( f \), which is a function of two variables \( x \) and \( y \). We will begin by differentiating \( f \) with respect to its variables.

Step 2: Determine the inner functions

The variables \( x \) and \( y \) are functions of \( r, s, \) and \( t \). Specifically, \( x = x(r, s, t) \) and \( y = y(r, s, t) \). These are the inner functions in our chain rule application.

Step 3: Apply the Chain Rule

The chain rule for a function \( u = f(x, y) \) with \( x = x(r, s, t) \) and \( y = y(r, s, t) \) is given by differentiating \( f \) with respect to each of its variables and then multiplying by the derivative of these variables. In mathematical terms:\[\frac{du}{dr} = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial r} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial r}\]\[\frac{du}{ds} = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial s}\]\[\frac{du}{dt} = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial t}\]

Step 4: Construct the Tree Diagram

In the tree diagram, \( u = f(x,y) \) is at the top, with branches to \( x \) and \( y \). Each \( x \) and \( y \) further branches out to \( r, s, \) and \( t \). \( \frac{\partial u}{\partial r}, \frac{\partial u}{\partial s}, \text{ and } \frac{\partial u}{\partial t} \) are computed along these paths using the derivatives outlined in the previous step.

Key Concepts

Partial Derivatives

When dealing with functions of multiple variables, like our function \( u = f(x, y) \), you might need to understand partial derivatives to differentiate effectively. A partial derivative is a derivative where we choose one variable to differentiate while keeping others constant.

In the context of the Chain Rule, we use partial derivatives to find how a function changes with respect to one variable, say \( x \), while treating the other variable, \( y \), as a constant. For example:

• \( \frac{\partial f}{\partial x} \) represents the rate of change of \( f \) with \( x \) while keeping \( y \) constant.
• \( \frac{\partial f}{\partial y} \) similarly defines how \( f \) changes as \( y \) varies, with \( x \) constant.

Understanding partial derivatives is crucial as they form the basis for applying the Chain Rule in multivariable calculus, allowing us to trace changes through nested functions.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions with more than one variable. In this exercise, the function \( u = f(x, y) \) is examined, where both \( x \) and \( y \) depend on the variables \( r, s, \) and \( t \).

This extension of single-variable calculus involves numerous powerful tools. The Chain Rule is one such tool, helping to manage the complexity by looking at how a change in one variable affects the outcome through a series of dependencies. It allows us to compute \( \frac{du}{dr} \), \( \frac{du}{ds} \), and \( \frac{du}{dt} \), reflecting changes in \( u \) as the variables \( r, s, \) or \( t \) change.

• These calculations illustrate the interconnected nature of multi-variate functions.
• It’s essential for problems involving physics, economics, and other fields where variables interact.

Getting a firm grip on multivariable calculus provides a deeper understanding of real-world systems where variables seldom operate in isolation.

Tree Diagram

Tree diagrams offer a visual way to understand the dependencies across functions which is particularly useful in applying the Chain Rule in multivariable calculus.

Starting with the function \( u = f(x, y) \), the tree diagram helps break down this function into its constituents. Here’s how it proceeds:

• Begin with \( u = f(x, y) \) at the top of the tree.
• Create branches towards \( x \) and \( y \), showing their dependencies.
• Further branch each of \( x \) and \( y \) into \( r, s, \) and \( t \), showcasing their functional relationships.
• Along these branches, note the partial derivatives \( \frac{\partial x}{\partial r} \), \( \frac{\partial y}{\partial s} \), etc.

This diagram systematically lays out the path needed to apply the Chain Rule, making it far easier to track and compute partial derivatives for complex problems. It's like a roadmap of the function, giving a clear depiction of the various pathways through which variable changes propagate.