sigmoid函数求导-只要四步_sigmoid函数求导过程

sigmoid函数的导数是$f^{\prime }(x)=f(x)(1-f(x))$
推导过程如下：

1.先将f(x)稍微变形
$f(x)=\frac{1}{1+e^{-x}}=\frac{e^x}{e^x+1}=1-{\left(e^x+1\right)}^{-1}$

2.求导：高等数学-符合求导法则
f ′ ( x ) = ( − 1 ) ( − 1 ) ( e x + 1 ) − 2 e x = ( 1 + e − x ) − 2 e − 2 x e x = ( 1 + e − x ) − 1 ⋅ e − x 1 + e − x = f ( x ) ( 1 − f ( x ) ) ​

$$\begin{aligned}f ^ { \prime } ( x )& = ( - 1 ) ( - 1 ) \left( e ^ { x } + 1 \right) ^ { - 2 } e ^ { x } \\ & = \left( 1 + e ^ { - x } \right) ^ { - 2 }e ^ { - 2 x } e ^ { x } \\ & = \left( 1 + e ^ { - x } \right) ^ { - 1 } \cdot \frac { e ^ { - x } } { 1 + e ^ { - x } } \\ & = f ( x ) ( 1 - f ( x ) ) \end{aligned}$$ ​ f′(x)​=(−1)(−1)(ex+1)−2ex=(1+e−x)−2e−2xex=(1+e−x)−1⋅1+e−xe−x​=f(x)(1−f(x))​​