Differentiating a polar function

[kgal]

Homework Statement

let z=f(x,y) be a differentiable function. If we change to polar coordinates, we make the substitution x=rcos(θ), y=rsin(θ), x^2+y^2=r^2 and tan(θ) = y/x.
a. Find expressions ∂z/∂r and ∂z/∂θ involving ∂z/∂x and ∂z/∂y.
b. Show that (∂z/∂x)^2 + (∂z/∂y)^2 = (∂z/∂r)^2 + (1/r^2)(∂z/∂θ)^2.

The Attempt at a Solution

a. i understand that f(x,y) in polar is f(r,θ) but don't understand how to calculate the partial derivatives of ∂z/∂x and ∂z/∂y because there is not know function for z...

 

[HallsofIvy]

Do you know the chain rule for functions of two variables?

 

[kgal]

in the specific case of this problem they come out like this:
∂z/∂r = (∂z/∂x)(∂x/∂r) + (∂z/∂y)(∂y/∂r)
∂z/∂θ = (∂z/∂x)(∂x/∂θ ) + (∂z/∂y)(∂y/∂θ)

right?