z=y^2f(xy,e^x)求二阶偏导数

z=y^2f(xy,e^x)求二阶偏导数

Z = y² F(xy，e^x) u=xy， v=e^x
Z = y² F(u，v)
Z'x = y² (F'u y + F'v e^x)
Z'x = y³ F'u + y²e^x F'v
Z''xx = y³(yF''uu +e^xF''uv)
= y^4 F''uu + y³e^x F''uv............(1)
Z''yy 和Z''xy与Z''xx求法类似！

z = y^2 f(xy，e^x) 设u=xy， v=e^x，则 z = y^2 f(u，v)。 求偏导数如下： (1) z'x = y^2 (f'u*y + f'v*e^x) z'x = y^3f'u + y^2e^x f'v z''xx = y^3(yf''uu +e^xf''uv)+y^2e^xf'v+y^2e^x(f''vu*y+f''vv*e^x) = y^4 f''uu + y^3e^x f''uv+y^2e^xf'v+y^3e^xf''vu+y^2e^2xf''vv =y^4 f''uu +2y^3e^x f''uv+y^2e^xf'v+y^2e^2xf''vv (2) z'y=2yf(u,v)+y^2(f'u*x+f'v*0)=2yf（u，v）+f'u*xy^2. z''yy =2f（u,v）+2y(f'u*x+f'v*0)+x*2y*f'u+xy^2*(f''uv*0+f''uu*e^x) =2f（u,v）+4xyf'u+xy^2e^xf''uu (3) z'x = y^3f'u + y^2e^x f'v z''xy =3y^2f'u+y^3*(f'uv*0+f'uu*x)+e^x*2yf'v+e^xy^2*(f'vv*0+f'vu*x) =3y^2f'u+xy^3f'uu+2e^x*yf'v+xy^2e^xf'vu

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