设u=f（x，y，z）有连续偏导数，y=y（x）和z=z（x）分布由方程exy-y=0和ez-xz=0所确定，求dudx．

Question

设u=f（x，y，z）有连续偏导数，y=y（x）和z=z（x）分布由方程e^xy-y=0和e^z-xz=0所确定，求(du)/(dx)．

Answer 1

∵u=f（x，y，z）有连续偏导数，y=y（x）和z=z（x）
∴(du)/(dx)＝(∂f)/(∂x)+(∂f)/(∂y)(dy)/(dx)+(∂f)/(∂z)(dz)/(dx)…①
又由e^xy-y=0，两边对x求导得：exy(y+x(dy)/(dx))−(dy)/(dx)＝0
∴(dy)/(dx)＝(yexy)/(1−xexy)=(y2)/(1−xy)
由e^z-xz=0，两边对x求导得：ez(dz)/(dx)−z−x(dz)/(dx)＝0
∴(dz)/(dx)＝(z)/(ez−x)＝(z)/(x(z−1))
∴代入①得：
(du)/(dx)＝(∂f)/(∂x)+(y2)/(1−xy)(∂f)/(∂y)+(z)/(x(z−))(∂f)/(∂z)