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Chapter 12: Problem 28

Use a tree diagram to write the required Chain Rule formula. $$\begin{aligned} &w=f(x, y, z), \text { where } x=g(t), y=h(s, t), \text { and } z=p(r, s, t)\\\ &\text { Find } \partial w / \partial t \end{aligned}$$

Short Answer

Expert verified
Based on the given information and the step-by-step solution provided, write the chain rule formula for \(\frac{\partial w}{\partial t}\), where \(w = f(x, y, z)\), \(x = g(t)\), \(y = h(s, t)\), and \(z = p(r, s, t)\). Answer: \(\frac{\partial w}{\partial t} = \frac{\partial w}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial w}{\partial y}\frac{\partial y}{\partial t}\)

Step by step solution

01

Draw the Tree Diagram for the Chain Rule

We'll start by setting up our tree diagram to display the relationships between the variables. Here's a visual representation of the tree: 
         w
        /|\
       / | \
      x  y  z
       \ / \_/ 
        t  s  r
- For \(w\), we have 3 branches: \(x = g(t), y = h(s, t),\) and \(z = p(r, s, t)\) - Each of these branches represents a function, and the variables of these functions (such as \(g(t), h(s, t),\) and \(p(r, s, t)\)) compose another level.
02

Determine Paths from \(w\) to \(t\) in the Tree Diagram

Next, we need to determine all the paths from \(w\) to \(t\): 1. \(w\) to \(x\) to \(t\): This path covers the derivative of \(w\) with respect to \(x\) and the derivative of \(x\) with respect to \(t\) 2. \(w\) to \(y\) to \(t\): This path covers the derivative of \(w\) with respect to \(y\) and the derivative of \(y\) with respect to \(t\) Now, we'll use these paths to create the required Chain Rule formula.
03

Write the Chain Rule Formula

Using all possible paths from \(w\) to \(t\), we will write out the chain rule formula for \(\frac{\partial w}{\partial t}\): $$\frac{\partial w}{\partial t} = \frac{\partial w}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial w}{\partial y}\frac{\partial y}{\partial t}$$ This formula takes into account both possible paths between \(w\) and \(t\). Keep in mind that we did not include the \(z = p(r, s, t)\) path since \(t\) is not present there.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Derivatives
Partial derivatives are essential tools in multivariable calculus. They allow us to measure the rate of change of a function with respect to one variable while keeping others constant. Consider a multivariable function like \(w=f(x, y, z)\). Here, \(x\), \(y\), and \(z\) are variables that affect \(w\) in different ways.
  • Notation: Partial derivatives are usually denoted by \(\frac{\partial w}{\partial x}\), \(\frac{\partial w}{\partial y}\), etc., agiating the variable you are differentiating with respect to.
  • Computation: To find \(\frac{\partial w}{\partial x}\), treat \(y\) and \(z\) as constants and differentiate \(w\) with respect to \(x\).
Understanding partial derivatives is crucial when working with functions of multiple variables, as they help analyze how each individual variable influences the function's behavior.
Tree Diagram
A tree diagram is a helpful visual tool that maps relationships between variables in a function. For the function \(w=f(x, y, z)\), where each variable is influenced by others like \(x=g(t)\), a tree diagram can simplify the chain rule application.
  • Structure: Start with \(w\) at the top. Draw branches to \(x\), \(y\), and \(z\). Each branch represents a relationship dictated by a function.
  • Purpose: It helps in identifying all paths between variables, relevant in applying the chain rule.
In our scenario, the diagram clarifies how changes in \(t\) affect \(w\) by showing direct and indirect influences, vital for calculating \(\frac{\partial w}{\partial t}\).
Functions of Multiple Variables
Functions of multiple variables like \(w=f(x, y, z)\) involve parameters representing different dimensions or factors affecting the outcome. Unlike single-variable functions, these can manage complex systems.
  • Description: They map an input set of variable combinations to a result, harnessing relationships within the function.
  • Application: Used extensively in physics, engineering, and economics to model real-world phenomena.
Grasping these functions involves understanding dependencies and dimensional influences. Partial derivatives and tree diagrams are tools helping to dissect this complexity, providing insight into individual contributions to the overall system.

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