Question

Use a tree diagram to write out the chain rule for the given case. Assume all functions are differentiable. U=f(s,t), where s=s(w,x,y,z), t=t(w,x,y,z)

Answer 1

Whew... I'll get you started...
\[ dU = \frac{\partial f}{\partial s}ds + \frac{\partial f}{\partial t} dt \]
where
\[ds = \frac{\partial s}{\partial w}dw + \frac{\partial s}{\partial x}dx + ...... \]
and likewise for dt.