シグモイド関数の微分
$$
\sigma ( x ) = \frac { 1 } { 1 + e ^ { – x } }
$$
$$
\frac { \partial } { \partial x } \sigma ( x ) = \sigma ( x ) ( 1 – \sigma ( x )
$$
計算
$$
\begin{aligned} \frac { \partial } { \partial x } \sigma ( x ) & = \frac { – \left( – e ^ { – x } \right) } { \left( 1 + e ^ { – x } \right) ^ { 2 } } \\ & = \frac { e ^ { – x } } { \left( 1 + e ^ { – x } \right) ^ { 2 } } \\ & = \left( \frac { 1 } { 1 + e ^ { – x } } \right) \frac { e ^ { – x } } { 1 + e ^ { – x } } \\ & = \sigma ( x ) \left( 1 – \frac { 1 } { 1 + e ^ { – x } } \right) \\ & = \sigma ( x ) ( 1 – \sigma ( x ) ) \end{aligned}
$$