sigmoid函数的导数为f’(x ) f'(x ) ) 1f ) x ) ) f ) x ) ) (1- f (x ) ) f ) ) x ) ) ) x ) )。
导出过程如下
1 .首先使f(x )稍微变形
f(x ) 1ex=exex1=1) ex1 ) 1f ) x ) ) FRAC({1e^{-x}}=(FRAC ) e ^ { x } { e ^ { x } {1}=1-left ()
2 .求导：遵循高等数学-求导规律
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))