sigmoid
sigmoid 函数是机器学习常用的一种激活函数，公式如下： \[ f(x) = \frac{1}{1+e^{-x}} \]
求导
\[\begin{equation} \begin{aligned} f(x)^{\prime} &= ((1+e^{-x})^{-1})^{\prime} \\ &= (-1)\times(1+e^{-x})^{-2} \times e^{-x} \times (-1) \\ &= (1+e^{-x})^{-2} \times e^{-x} \\ &= \frac{e^{-x}}{(1+e^{-x})^2} \\ &= \frac{1+e^{-x}-1}{(1+e^{-x})^2} \\ &= \frac{1+e^{-x}}{(1+e^{-2})^2} - \frac{1}{(1+e^{-x})^2} \\ &= \frac{1}{1+e^{-x}} - \frac{1}{(1+e^{-x})^2} \\ &= \frac{1}{1+e^{-x}}(1-\frac{1}{1+e^{-x}}) \\ &= f(x)(1-f(x)) \end{aligned} \end{equation}\]