Question

e^x-xyz=0，求∂^2z/∂x^2

e^x-xyz=0，求∂^2z/∂x^2
课本答案是(2y^2ze^x-2xy^3z-y^2z^2e^x)/(e^x-xy)^3，我感觉不对诶

Answer 1

e^xdx = xydz + xzdy + yzdx
(e^x - yz)dx - xzdy = xydz
dz = (e^x - yz)/xy * dx - z/y * dy
则∂z/∂x = (e^x - yz)/xy
∂²z/∂x² = ∂(∂z/∂x)/∂x = ∂[(e^x - yz)/xy]/∂x
= [(e^x - y∂z/∂x)xy - y(e^x - yz)]/x²y²
= [(x-2)e^x + 2yz]/x²y
= (x²-2x+2)e^x/x³y
后面那个结果和用z = e^x/xy对x求偏导数结果一样