Question

求偏导数z=f(xy,x^2+y^2)具有二阶连续偏导数

求a^2z/ax^2 a^2z/axay

Answer 1

设f1=fxy(xy,x^2+y^2),f2=fx^2+y^2(xy,x^2+y^2)
az/ax=f1*y+f2*2x
a^2z/ax^2=f11*y^2+f12*2xy+f21*2xy+f22*4x^2
a^2z/axay=f11*xy+f12*2y^2+f1+f21*2x^2+f22*4xy

Answer 2

Dz/Dx = f1*y + f2*(2x) = y*f1 + 2x*f2，
D²z/Dx² = (D/Dx)(y*f1 + 2x*f2) = [y*(y*f11 + 2x*f12)+ 2*f2 + 2x*(y*f21 +2x*f22)] = ……，
D²z/DxDy = (D/Dy)(y*f1 +2x*f2) = [f1 + y*(x*f11 + 2y*f12)+ 2x*(x*f21 +2y*f22)] = ……。