绝对值函数的二阶导数，指数函数（幂为绝对值函数）的二阶导数，delta函数

rayyy9

已于 2024-12-26 07:21:04 修改

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分类专栏： calculus 文章标签：人工智能算法

于 2024-12-26 07:20:46 首次发布

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绝对值函数的二阶导数的计算过程如下

$\frac{\partial^2 \left | x \right |}{\partial x^2}$

x>0， $\frac{\partial^2 \left | x \right |}{\partial x^2}=\frac{\partial^2 x}{\partial x^2}=0$

x<0， $\frac{\partial^2 \left | x \right |}{\partial x^2}=\frac{\partial^2 (-x)}{\partial x^2}=0$

x=0呢

$\int_{0^{-}}^{0^{+}}\frac{\partial^2 \left | x \right |}{\partial x^2}dx =\int_{0^{-}}^{0^{+}}d(\frac{\partial |x|}{\partial x})=1-(-1)=2$

$\int_{0^{-}}^{0^{+}}\delta \left ( x \right )dx =1$

所以 $\frac{\partial^2 \left | x \right |}{\partial x^2}=2\delta \left ( x \right )$

$\frac{\partial^2 exp\left ( \left | x \right | \right )}{\partial x^2}$

x>0, $\frac{\partial^2 exp\left ( \left | x \right | \right )}{\partial x^2}=\frac{\partial^2 exp\left ( x \right )}{\partial x^2}=exp\left ( x \right )$

x<0, $\frac{\partial^2 exp\left ( \left | x \right | \right )}{\partial x^2}=\frac{\partial^2 exp\left ( -x \right )}{\partial x^2}=exp\left ( -x \right )$

$\int_{0^{-}}^{0^{+}}\frac{\partial^2 exp(\left | x \right |)}{\partial x^2}dx =\int_{0^{-}}^{0^{+}}d(\frac{\partial exp(|x|)}{\partial x})=exp(0)-(-exp(0))=2$