1. a) Calculate the partial derivatives for and prove that
b)If and , calculate using the chain rule.
c)For where use the chain rule to calculate , and .
I am having a lot of trouble with this problem.
a) To find the partial derivative of f(x,y) with respect to x, treat y as a constant, and vice versa. Have you tried this?
b) Maybe someone else can help me out here because I don't see how the chain rule applies. Replace all instances of x with x(u) = u^2 and likewise for y in f(x,y). You are left with a univariate problem of finding the derivative of f(u) with respect to u. (Here we are overloading the variable name f as in: f(u) = f(x(u),y(u)), which may be confusing, but that seems to be the notation of the problem.)
c) First of all, g(r) simplifies to . For the first partial derivative, you'll solve exactly as you would
d/dx (x^2 + c)^2
As in,
d/dx (x^2 + c)^2 = 2*(x^2 + c) * 2x
Hopefully it makes more sense now?