Problem 19 - 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. $$w=f(r, s, t), \quad \text { where } r=r(x, y), \quad s=s(x, y), \quad t=t(x, y)$$

Step-by-Step Solution

Answer

Use the chain rule to express \(\frac{\partial w}{\partial x}\) and \(\frac{\partial w}{\partial y}\).

Step 1: Identify the Overall Structure

We start with the composite function \(w=f(r, s, t)\), where \(r, s, t\) are functions of \(x\) and \(y\). We need to express the derivative \(\frac{\partial w}{\partial x}\) and \(\frac{\partial w}{\partial y}\) using the chain rule, mapping out the dependencies using a tree diagram.

Step 2: Construct the Tree Diagram

In the tree diagram, the function \(w\) is at the top with branches leading to \(r\), \(s\), and \(t\). Each of these branches further splits since \(r\), \(s\), and \(t\) each depend on \(x\) and \(y\). Thus, \(r\) has branches to \(x\) and \(y\), \(s\) to \(x\) and \(y\), and \(t\) to \(x\) and \(y\).

Step 3: Apply the Chain Rule for \(\frac{\partial w}{\partial x}\)

Using the tree diagram, express \(\frac{\partial w}{\partial x}\) by accounting for each path from \(w\) to \(x\). Use the chain rule:\[\frac{\partial w}{\partial x} = \frac{\partial f}{\partial r} \cdot \frac{\partial r}{\partial x} + \frac{\partial f}{\partial s} \cdot \frac{\partial s}{\partial x} + \frac{\partial f}{\partial t} \cdot \frac{\partial t}{\partial x}.\]

Step 4: Apply the Chain Rule for \(\frac{\partial w}{\partial y}\)

Similarly, using the same paths from \(w\) to \(y\), express \(\frac{\partial w}{\partial y}\) with the chain rule:\[\frac{\partial w}{\partial y} = \frac{\partial f}{\partial r} \cdot \frac{\partial r}{\partial y} + \frac{\partial f}{\partial s} \cdot \frac{\partial s}{\partial y} + \frac{\partial f}{\partial t} \cdot \frac{\partial t}{\partial y}.\]

Key Concepts

Tree Diagram

A tree diagram is a useful visual tool, especially in understanding the dependencies between different variables in composite functions. When dealing with functions where one function is nested within others, the tree diagram helps map out the different layers and show how each variable is interconnected.
For example, if you consider the function from the exercise, where \(w = f(r, s, t)\) and each of \(r, s,\) and \(t\) depend on \(x\) and \(y\), the tree diagram starts with \(w\) at the top level.
Then, three branches emerge from \(w\) pointing to \(r, s,\) and \(t\). Further, each of these branches split again since \(r, s,\) and \(t\) depend on \(x\) and \(y\).
This breaks down the complex relationship into understandable parts, aiding in applying the chain rule for derivatives by clearly showing each variable's dependence.

Partial Derivative

Partial derivatives are important when dealing with functions of multiple variables. They give the rate at which the function changes as one of the variables changes, while the others remain constant.
This is crucial in finding derivatives of composite functions, like in the exercise's case with \(w = f(r, s, t)\).
Here, the goal is to determine how \(w\) changes with respect to \(x\) and \(y\) by keeping track of its dependence on \(r, s,\) and \(t\).
The notation \(\frac{\partial w}{\partial x}\) means you're looking at the rate of change of \(w\) with changes in \(x\), ignoring any direct changes in \(y\), and similarly for \(\frac{\partial w}{\partial y}\).
By using partial derivatives and summing the effects via the chain rule, we get a complete understanding of the different influences \(x\) or \(y\) have on \(w\).

Composite Function

Composite functions occur when the output of one function becomes the input of another. In this case, \(w = f(r, s, t)\) represents a composite function, with \(r = r(x, y), s = s(x, y), t = t(x, y)\).
This adds layers to the equation, where changes in \(x\) or \(y\) impact \(r, s,\) and \(t\), which in turn affect \(w\).
The key to managing composite functions is to understand how each function feeds into the next, influencing overall changes.

• First, know each variable's dependence.
• Second, use tools like tree diagrams to help visualize these dependencies.
• Third, apply the chain rule to derive the step-by-step changes with respect to desired variables.

This ensures that all factors of change are considered, making it easier to compute derivatives accurately.

Differentiable Functions

When applying the chain rule to composite functions, it's important to ensure that all the involved functions, like \(f, r, s,\) and \(t\) in our example, are differentiable.
A function is differentiable if it has a derivative at every point in its domain, meaning it's smooth without any sharp turns or cusps.
This smoothness allows us to calculate derivatives systematically using the chain rule.
The assurance of differentiability simplifies calculations because you can assume small incremental changes in one variable linearly affect another, making the mathematical operations straightforward.
Thus, ensuring differentiability is crucial, as it forms the backbone of utilizing derivatives effectively throughout problem-solving in calculus.