Suppose that you are given an equation of the form F(x, y, z) = 0, for example,...

Suppose that you are given an equation of the form

F(x, y, z) = 0,

for example, something like x³z + y cos z + (sin y)/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

(b) Use part (a) to find ∂z/∂x and ∂z/∂y where z is given by the equation xyz = 2. Check your result by explicitly solving for z and then calculating the partial derivatives.