Econ 11B Mathematics for Econ - Chain Rule Problem






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). 


Start with your original function;

            

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.
































 



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