Question

求偏导数设函数f(x,y)={xy(x^2-y^2)/(x^2+y^2) ,当(x,y) ≠(0,0);0,当（x,y）

Answer 1

先求函数的全导数为：

df(x,y)={[xy(x^2-y^2)]'(x^2+y^2)-xy(x^2-y^2)(x^2+y^2)'}/(x^2+y^2)^2

={[(xy)'(x^2-y^2)+(xy)(x^2-y^2)'](x^2+y^2)-xy(x^2-y^2)(2xdx+2ydy)}/(x^2+y^2)^2

={[(ydx+xdy)(x^2-y^2)+xy(2xdx-2ydy)](x^2+y^2)-2xy(x^2-y^2)(xdx+ydy)}/(x^2+y^2)^2

=【y(x^4-4x^2y^2-y^4)/(x^2+y^2)^2】dx-【x(x^4-5y^4)/(x^2+y^2)^2】dy

前者dx前面的为对x的偏导数,后于dy前面的为对y的偏导数.