### Econ 11B Mathematics for Econ - Chain Rule Problem

 Question 5 -  Let ,  where; Using the Chain Rule, find the partial derivative . Setting up the Chain Rule; Now, for each one of those  (∂ #  /  ∂ #) terms; Plugging in each of these; Simplifying this delivers the following solution (and alternate forms); Alternatively, you could have solved this problem using substitution. Sometimes the chain rule is easier, sometimes substitution is easier (actually, I generally think the chain rule is easiest, since you also get all those first order conditions in a handy matrix there too... but whatever). Now plug in each of those equations that we were given for x, y & w. Which implies;  And we know what ...w(r,s), x(r,s) & y(r,s) are ..all equal, let’s plus those in; Also, you could solve this problem in terms of the variables  r & s.
What do this? notice that our solutions above are all in terms of y, x, w, r & s -- but we have equations for y, x, & w in terms of r & s. So this would be a further simplification. 