4.隐函数的偏导数与全微分【15】设u=f(x,y,z),z=z(x,y)是由方程φ(x+y,z)=1所确定的隐函数，其中f和φ有二阶连续偏导数且φ₂≠0，则(partial^2u)/(partial xpartial y)=____.

由 $\phi(x+y,z)=1$，得 $\frac{\partial z}{\partial x} = -\frac{\phi_1'}{\phi_2'}$，$\frac{\partial z}{\partial y} = -\frac{\phi_1'}{\phi_2'}$。 对 $u = f(x,y,z)$ 求导： \[ \frac{\partial u}{\partial x} = f_1' - f_3' \cdot \frac{\phi_1'}{\phi_2'} \] 再对 $y$ 求导： \[ \frac{\partial^2 u}{\partial x \partial y} = f_{12}'' - \frac{f_{13}'' \phi_1' + f_{32}'' \phi_1' + f_3' \phi_{11}''}{\phi_2'} + \frac{f_{33}'' \phi_1'^2 + f_3' \phi_1' (\phi_{12}'' + \phi_{21}'')}{\phi_2'^2} \] \[ \boxed{f_{12}'' - \frac{f_{13}'' \phi_1' + f_{32}'' \phi_1' + f_3' \phi_{11}''}{\phi_2'} + \frac{f_{33}'' \phi_1'^2 + f_3' \phi_1' (\phi_{12}'' + \phi_{21}'')}{\phi_2'^2}} \]