Question

Z=f(u,x,y),u=xe^y,其中f具有二阶连续偏导数，求Z〃xx

求详解

Answer 1

δz/δx=f1(u,x,y)e^y+f2(u,x,y),
δz/δy=f1(u,x,y)xe^y+f3(u,x,y),

δ^2z/δx^2=[f11(u,x,y)e^y+f12(u,x,y)]e^y+
+f12(u,x,y)e^y+f22(u,x,y),
δ^2z/δxδy=[f11(u,x,y)xe^y+f13(u,x,y)]e^y+f1(u,x,y)e^y
+f12(u,x,y)xe^y+f23(u,x,y),
δ^2z/δy^2=[f11(u,x,y)xe^y+f13(u,x,y)]xe^y+f1(u,x,y)xe^y
+f13(u,x,y)xe^y+f33(u,x,y).

Answer 2

Z'x=f'u(u,x,y)*u'x+f'x(u,x,y)
Z''xx=f''u(u,x,y)*u'x*u'x+f'u(u,x,y)*(u''x)+f''x(u,x,y)