sigmoid函数求导

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

1.先将f(x)稍微变形 f ( x ) = 1 1 + e − x = e x e x + 1 = 1 − ( e x + 1 ) − 1 ​ f ( x ) = \frac { 1 } { 1 + e ^ { - x } }=\frac { e ^ { x } } { e ^ { x } + 1 } = 1 - \left( e ^ { x } + 1 \right) ^ { - 1 }​ f(x)=1+e−x1​=ex+1ex​=1−(ex+1)−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))​​