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The dependency diagram is a valuable tool for understanding how variables influence each other in a function. In the context of the given exercise, the variable \( z \) is a function that directly depends on three other variables: \( u, v, \) and \( w \). Each of these intermediate variables in turn depends on a common base variable, \( t \). To visualize this:

• Place \( z \) at the top of the diagram as it is the outcome we want to understand the change of.
• In the middle layer, include \( u, v, \) and \( w \) because they are directly affecting \( z \).
• At the bottom, you have \( t \) connecting to each of \( u, v, \) and \( w \), indicating it affects all three.

This hierarchical flow from \( t \) to \( z \) helps illustrate how changes in \( t \) ripple through \( u, v, \) and \( w \) to ultimately affect \( z \). This diagram is crucial before applying the Chain Rule because it lays out all the necessary dependencies.

Partial derivatives allow us to explore how small changes in one variable affect a function when all other variables are held constant. For the function \( z = f(u, v, w) \), we consider partial derivatives with respect to \( u, v, \) and \( w \). When we compute \( \frac{\partial z}{\partial u} \), we are essentially seeing how \( z \) changes if we alter \( u \) but keep \( v \) and \( w \) unchanged. Similarly:

• \( \frac{\partial z}{\partial v} \) shows the sensitivity of \( z \) to changes in \( v \).
• \( \frac{\partial z}{\partial w} \) provides insight into how changes in \( w \) influence \( z \).

These derivatives are foundational in setting up the Chain Rule for the composed function \( z = f(u, v, w) \) since they quantify how each path affects \( z \).

Composed functions occur when a function is made up of other functions. In this case, \( z = f(u, v, w) \) is a composition of:\( u = g(t) \), \( v = h(t) \), and \( w = k(t) \). This setup implies that \( z \) is indirectly influenced by \( t \) through its changing inputs \( u, v, \) and \( w \). Understanding composed functions allows us to see how multiple dependencies interact. In particular:

• \( z \) changes with alterations in \( u \), \( v \), and \( w \) as introduced by \( t \).
• The composition here is crucial because each intermediate function (\( g(t), h(t), \) and \( k(t) \)) directly interface with the base variable \( t \).

In composed functions, it’s essential to consider where each part of the function originates from to accurately apply calculus operations such as the Chain Rule.

The derivative with respect to time, \( \frac{dz}{dt} \), captures how the function \( z \) changes as time \( t \) changes. To resolve this in composed functions using the Chain Rule, we follow the influence paths illustrated in the dependency diagram. For \( z = f(u, v, w) \) where each of \( u, v, w \) is a function of \( t \), the Chain Rule breaks down the contribution to \( z \)'s rate of change from each variable:

• First, express the sensitivity of \( z \) to \( u, v, \) and \( w \) via partial derivatives \( \frac{\partial z}{\partial u} \), \( \frac{\partial z}{\partial v} \), and \( \frac{\partial z}{\partial w} \).
• Multiply each by the respective rate of change of \( u, v, \) and \( w \) with respect to \( t \), given by \( \frac{du}{dt} \), \( \frac{dv}{dt} \), and \( \frac{dw}{dt} \).

Thus, \( \frac{dz}{dt} = \frac{\partial f}{\partial u} \cdot g'(t) + \frac{\partial f}{\partial v} \cdot h'(t) + \frac{\partial f}{\partial w} \cdot k'(t) \), representing how every variable depending on \( t \) contributes to the overall change in \( z \). This calculation is pivotal in many real-world scenarios where time is a constant factor influencing many dynamic systems.