Question

一个求导的题求解

设f(x-y,xy)=x³-y³则f′(x,y)=?
请写出具体步骤谢谢啦
我想求的是导数。我看到这道题目书上给的答案是3（x²+y）可是我不知道推导出来的步骤。楼下的朋友谢谢你拉。能给我个具体的步骤吗？

Answer 1

设u=x-y, v=xy
x³-y³=(x-y)(x²+xy+y²)=(x-y)[(x-y)²+3xy]=u(u²+3v)
即f(u,v)=u(u²+3v)

∂f/∂x=(∂f/∂u)*(∂u/∂x)+(∂f/∂v)*(∂v/∂x)
=(3u²+3v)+3u*y
=3(x-y)²+3xy+3xy-3y²
=3x²

∂f/∂y=(∂f/∂u)*(∂u/∂y)+(∂f/∂v)*(∂v/∂y)
=-(3u²+3v)+3u*x
=-3(x-y)²-3xy+3x²-3xy
=-3y²

f的全微分
df=(∂f/∂x)dx+(∂f/∂y)dy=3x²dx-3y²dy