##### 高阶偏导数 - 高阶偏导数 - 以二元函数 $f(x,y)$ 为例, 如果存在两个[[偏导数]] $\displaystyle\frac{\partial{z}}{\partial{x}}=f_x(x,y)$, $\displaystyle\frac{\partial{z}}{\partial{y}}=f_y(x,y)$, 并且这两个偏导函数也存在偏导数, 则称为二阶偏导数, 高阶类似. 对变量求导次序的不同有四个二阶偏导数, 其中第二三两个偏导数称为混合偏导数 - $\displaystyle\frac{\partial}{\partial{x}}(\frac{\partial{z}}{\partial{x}})=\frac{\partial^2{z}}{\partial{x^2}}=f_{xx}(x,y)$ - $\displaystyle\frac{\partial}{\partial{y}}(\frac{\partial{z}}{\partial{x}})=\frac{\partial^2{z}}{\partial{x}\partial{y}}=f_{xy}(x,y)$ - $\displaystyle\frac{\partial}{\partial{x}}(\frac{\partial{z}}{\partial{y}})=\frac{\partial^2{z}}{\partial{y}\partial{x}}=f_{yx}(x,y)$ - $\displaystyle\frac{\partial}{\partial{y}}(\frac{\partial{z}}{\partial{y}})=\frac{\partial^2{z}}{\partial{y^2}}=f_{yy}(x,y)$