I've saved this problem for last, because it is an application of the Chain Rule. I've mentioned in class that the Chain Rule in Multivariable Calculus is used more as a theoretical tool rather than a computational one. The results of this problem are used in the very last week of class. Parts (b) and (c) are extra credit. Let z = g(x,y). Let R(x, y, z) be a function composed with z = g(x,y), so your function will look like R(x,y,g(x,y)).
(a) [3 points] Find ∂R/∂x and ∂R/∂y.
(b) [2 points] Let Q(x,y, z) = Q(x,y,g(x,y)) and P(x,y,z) = P(x, y, g(x,y)) (same function g). Find the following, using properties of derivatives and the chain rule: ∂/∂x(Q + R ∂g/∂y) and ∂/∂y(P + R ∂g/∂x) and identify properties when you use them.
(c) [2 points] Using the previous parts, show that ∂/∂x(Q + R ∂g/∂y) - ∂/∂y(P + R ∂g/∂x) = (-∂g/∂x)(∂R/∂y - ∂Q/∂z) - ∂g/∂y(∂P/∂z - ∂R/∂x) + (∂Q/∂x - ∂P/∂y) and identify properties when you use them.