Use the Chain Rule to find the indicated partial derivatives. w = xy + yz + zx, x = r cos(θ), y = r sin(θ), z = rθ;
r=4 theta=pi/2
dw/dr=
dw/dtheta=

By the chain rule,

and similarly, is the same with replaced with .

We have

Putting everything together, we get

Replace with their equivalent expressions in terms of :

When and , the derivative has a value of

The same process will yield

Yes. Juan will break even because 16 and -16 are a zero pair. -16 is the additive inverse of 16. So the sum will be 0.