Question

请问这个多元函数求导的题目怎么做？

求详细步骤和答案

Answer 1

如图所示

Answer 2

Answer 3

令 u = xy， v = x+y， 则 z = (1/x)f(u) + yf(v)
∂z/∂x = (-1/x^2)f(u) + (1/x)(df/du)(∂u/∂x) + y(df/dv)(∂v/∂x)
= (-1/x^2)f(u) + (y/x)(df/du) + y(df/dv)
∂^2z/∂x∂y = (-1/x^2)(df/du)(∂u/∂y) + (1/x)(df/du) + (y/x)(d^2f/du^2)(∂u/∂y)
+ df/dv + y(d^2f/dv^2)(∂v/∂y)
= (-1/x)(df/du) + (1/x)(df/du) + y(d^2f/du^2) + df/dv + y(d^2f/dv^2)
= y(d^2f/du^2) + df/dv + y(d^2f/dv^2)