二元复合函数求二阶偏导

二元复合函数求二阶偏导

$$设z=F(x+f(2x-y),y),其中F,f二阶连续偏导数，求\frac{{\partial }^{2}z}{\partial y^2}.$$

解：
$$\frac{\partial z}{\partial y}=F_1^{\prime }{f}^{\prime }\cdot (-1)+F_2^{\prime }$$
$$\frac{{\partial }^{2}z}{\partial y^2}=\frac{\partial F_1^{\prime }{f}^{\prime }\cdot (-1)}{\partial y}+\frac{\partial F_2^{\prime }}{\partial y}$$
其中：
$$\frac{\partial F_1^{\prime }{f}^{\prime }\cdot (-1)}{\partial y}=f^{\prime }(F_{11}^{\prime \prime }{f}^{\prime }-F_{12}^{\prime \prime })+F_1^{\prime }{f}^{\prime \prime }$$
$$\frac{\partial F_2^{\prime }}{\partial y}=F_{21}^{\prime \prime }{f}^{\prime }\cdot (-1)+F_{22}^{\prime \prime }$$
综上：
$$\frac{{\partial }^{2}z}{\partial y^2}=F_{11}^{\prime \prime }(f^{\prime }{)}^2-2F_{12}^{\prime \prime }{f}^{\prime }+F_1^{\prime }{f}^{\prime \prime }+F_{22}^{\prime \prime }$$