高数题，偏导数

Question

高数题，偏导数四道题，要详细过程，题目见图

Answer 1

令 u = xz，v = z-y，则 z = f(u,v), 两边对 x 求偏导，得
∂z/∂x = (∂f/∂u)(∂u/∂x) + (∂f/∂v)(∂v/∂x)
= (∂f/∂u)(z+x∂z/∂x) + (∂f/∂v)(∂z/∂x)
(1-x∂f/∂u-∂f/∂v)∂z/∂x = z∂f/∂u
∂z/∂x = z(∂f/∂u)/(1-x∂f/∂u-∂f/∂v)；
z = f(u,v), 两边对 y 求偏导，得
∂z/∂y = (∂f/∂u)(∂u/∂y) + (∂f/∂v)(∂v/∂y)
= (∂f/∂u)(x∂z/∂y) + (∂f/∂v)(∂z/∂y-1)
(1-x∂f/∂u-∂f/∂v)∂z/∂y = -∂f/∂v
∂z/∂y = (-∂f/∂v)/(1-x∂f/∂u-∂f/∂v).

Answer 2

不难，都是基础题。追答

先来一道练练手

追问

后面三题？