91Ó°ÊÓ

First, we need to draw a tree diagram to represent the dependencies between \(u\), \(v\), \(w\), \(x\), \(y\), and \(z\). u | v /|\ w x y | | | zmmmmm / \ t z The arrows represent the partial derivatives. For example, \(\frac{\partial{v}}{\partial{w}}\), \(\frac{\partial{v}}{\partial{x}}\) and \(\frac{\partial{v}}{\partial{y}}\).

We will follow the tree diagram to write the Chain Rule formula for \(\frac{\partial{u}}{\partial{z}}\). From the tree diagram we can find three different paths between \(u\) and \(z\): 1. \(u \rightarrow v \rightarrow w \rightarrow z\) 2. \(u \rightarrow v \rightarrow x \rightarrow t \rightarrow z\) 3. \(u \rightarrow v \rightarrow x \rightarrow z\) 4. \(u \rightarrow v \rightarrow y \rightarrow t \rightarrow z\) 5. \(u \rightarrow v \rightarrow y \rightarrow z\) Applying the Chain Rule, the partial derivative \(\frac{\partial{u}}{\partial{z}}\) is the sum of multiplying the derivatives along each path: \(\frac{\partial{u}}{\partial{z}} = \frac{\partial{u}}{\partial{v}}\frac{\partial{v}}{\partial{w}}\frac{\partial{w}}{\partial{z}} + \frac{\partial{u}}{\partial{v}}\frac{\partial{v}}{\partial{x}}\frac{\partial{x}}{\partial{t}}\frac{\partial{t}}{\partial{z}} + \frac{\partial{u}}{\partial{v}}\frac{\partial{v}}{\partial{x}}\frac{\partial{x}}{\partial{z}} + \frac{\partial{u}}{\partial{v}}\frac{\partial{v}}{\partial{y}}\frac{\partial{y}}{\partial{t}}\frac{\partial{t}}{\partial{z}} + \frac{\partial{u}}{\partial{v}}\frac{\partial{v}}{\partial{y}}\frac{\partial{y}}{\partial{z}}.\) Since we are asked to find the Chain Rule formula only, we can stop here. In case the given functions were provided, we would have to compute the relevant partial derivatives and substitute them into the formula to obtain the final result.