The law of derivative with implicit function: let f = e z-XYZ, then FX (derivative of f to x) =-yz, FZ (derivative of f to z) = e z-xy.

Δ Z/Δ X =-FX/FZ = YZ/(EZ-XY), it must be noted that Z in the first derivative is a function of X and Y.

Derive the first-order partial derivative by the derivative rule of quotient, and then

(δ^2z)/δx^2={y(δz/δx)(e^z-xy)-yz[(e^z)(δz/δx)-y]}/[(e^z-xy)^2)，

Δ z/Δ x = yz/(Ez-xy) is substituted into the above formula to get (Δ 2z)/Δ x 2,

The third analytic function of the second sub-question is unclear, so it is better to use mathtype and turn it into a graph to write a clear expression. The method can still be clearly stated: (1) First, take the derivative rule of implicit function from the equation x+y+z-xyz=0, and get the derivative function of implicit function Z to X determined by this equation; (2) Then take f (x, y, Z)= e(3) and substitute the derivative function of z to x obtained in the first step into the derivative function of z to x obtained in the second step, and then substitute it into the point (0, 1,-1) to obtain f'(0, 1,-/).