Derivative of Sigmoid Function

Sigmoid Function

${\color{Magenta} g(z)=\frac{1}{1+e^{-z}}}$

$\begin{align*} \frac{d}{dz}g(z) &=\frac{d}{dz}\frac{1}{1+e^{-z}} \\ &=-\frac{1}{\left ( 1+e^{-z} \right )^{2}}\frac{d}{dz}\left ( 1+e^{-z} \right )\\ &=-\frac{1}{({1+e^{-z}})^2}\left ( -e^{-z} \right )\\ &= \frac{1+e^{-z}-1}{\left ( 1+e^{-z} \right )^{2}}\\ &=\frac{1}{1+e^{-z}}\left ( 1-\frac{1}{1+e^{-z}} \right )\\ &=g(z)[1-g(z)] \end{align*}$

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Derivative of Sigmoid Function

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