Multivariable Calculus: Finding g'(0)

[theBEAST]

Homework Statement

Here is the question with the solution:
http://dl.dropbox.com/u/64325990/MATH%20253/midterm%202.PNG

I don't understand how to do a. How did they know to use x=0 and y=0 for Fx(0,0) and Fy(0,0)?

I understand that this is implicit differentiation using Chain Rule where
dy/dx = -Fx/Fy

But g'(0) only tells us that x=0 but tells us nothing about y. So how would they know to use y=0 as well?Edit: Also does anyone have an idea how to start part b g''(0)?

 

[cjc0117]

[STRIKE]Plug x=0 into F(x,g(x))=2 and solve for y=g(x).[/STRIKE]
EDIT: Actually, you can tell just from the fact that we let y=g(x) for the funtion F(x,y), and we are told that g(0)=0. y=g(x) is dependent on x. And when x=0, y=g(0), which we are told is equal to zero.