91Ó°ÊÓ

In this exercise, we have w as a dependent variable which is a function of x, y, and z. Meanwhile, x, y, and z are functions of r and s, which are the independent variables.

To create a tree diagram for this problem, we will start by placing the dependent variable w at the top of the tree. Then, we will branch out to the variables that w directly depends on, which are x, y, and z. Next, we need to branch out from x, y, and z to the independent variables r and s. So, we will create two branches from each of x, y, and z, leading to r and s.

Now that we have the tree diagram's structure in place, we will label the branches connecting w to x, y, and z with their corresponding partial derivatives: \(\frac{\partial w}{\partial x}\), \(\frac{\partial w}{\partial y}\), and \(\frac{\partial w}{\partial z}\). Similarly, we will label the branches connecting x, y, and z to r and s with their appropriate partial derivatives: - From x: \(\frac{\partial x}{\partial r}\) and \(\frac{\partial x}{\partial s}\). - From y: \(\frac{\partial y}{\partial r}\) and \(\frac{\partial y}{\partial s}\). - From z: \(\frac{\partial z}{\partial r}\) and \(\frac{\partial z}{\partial s}\). Now we have a complete Chain Rule tree diagram with branches labeled with the appropriate derivatives. The tree diagram looks like: w /|\ / | \ / | \ x y z /| | | |\ / | | | | \ r s r s r s And the labeled tree diagram is: w / | \ ∂w/∂x ∂w/∂y ∂w/∂z / | \ x y z /| | | \ ∂x/∂r ∂x/∂s | ∂z/∂r ∂z/∂s | / \ ∂y/∂r ∂y/∂s This tree diagram represents the Chain Rule relationships between w, x, y, z, r, and s, with branches labeled with appropriate derivatives.