Suppose that you are given an equation of the form:

F(x,y,z)=0

for example, something like x^3z+ycos(z)+(siny)/z=0. Then we may consider z to be defined implicitly as a function z(x,y).

a) Use the chain rule to show that if F and z(x,y) are both assumed to be differentiable, then

dz/dx = - Fx(x,y,z)/Fz(x,y,z) and dz/dy = -Fy(x,y,z)/Fz(x,y,z)

note: the letters after the F's are subscripts