1. **State the problem:** We want to find the derivative of $w=f(t)$ where $t=(x,y,z)$ and each of $x,y,z$ depends on variables $u,v$. Specifically, we want to find $\frac{\partial w}{\partial u}$ using the chain rule. 2. **Recall the chain rule formula:** For a function $w=f(x,y,z)$ where $x,y,z$ depend on $u,v$, the partial derivative of $w$ with respect to $u$ is given by: $$\frac{\partial w}{\partial u} = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial u} + \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial u} + \frac{\partial w}{\partial z} \cdot \frac{\partial z}{\partial u}$$ 3. **Explain the tree diagram approach:** - Start from $w$ at the top. - Branch down to $x,y,z$. - From each of $x,y,z$, branch down to $u,v$. - The derivative $\frac{\partial w}{\partial u}$ is the sum of all paths from $w$ to $u$ multiplying derivatives along the path. 4. **Write the formula explicitly:** $$\frac{\partial w}{\partial u} = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial u} + \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial u} + \frac{\partial w}{\partial z} \cdot \frac{\partial z}{\partial u}$$ 5. **Summary:** The chain rule for $w=f(t)$ with $t=(x,y,z)$ and $x,y,z$ functions of $u,v$ gives the derivative of $w$ with respect to $u$ as the sum of partial derivatives of $w$ with respect to each variable times the derivative of that variable with respect to $u$. This completes the chain rule formula using the tree diagram method.