Derivative of Sigmoid ($\sigma$)

Derivative of sigmoid or \sigma . It is used to gradient decent minimal of cost function.

$sigmoid(x) = s(x) = {1 \over 1-e^{-x}}$ $\frac d{dx}s(x) = \frac {d }{dx}\bigl({1 \over 1-e^{-x}}\bigl)$ $= \bigl( {1 \over 1+e^{-x}} \bigl)^2 \frac {d }{dx}\bigl({1 -e^{-x}}\bigl) \text{quotient rule}$ $= \bigl( {1 \over 1+e^{-x}} \bigl)^2 e^{-x} (-1)$ $= \bigl( {1 \over 1+e^{-x}} \bigl)\bigl( {1 \over 1+e^{-x}} \bigl)( -e^{-x})$ $= \bigl( {1 \over 1+e^{-x}} \bigl)\bigl( { -e^{-x} \over 1+e^{-x}} \bigl)$ $= s(x)(1-s(x))$