# Derivative of Sigmoid Let's denote the sigmoid function as $\sigma(x)=\frac{1}{1+e^{-x}}$. Then: $ \begin{aligned} \frac{d}{d x} \sigma(x) & =\frac{d}{d x}\left[\frac{1}{1+e^{-x}}\right] \\ & =\frac{d}{d x}\left(1+\mathrm{e}^{-x}\right)^{-1} \\ & =-\left(1+e^{-x}\right)^{-2}\left(-e^{-x}\right) \\ & =\frac{e^{-x}}{\left(1+e^{-x}\right)^2} \\ & =\frac{1}{1+e^{-x}} \cdot \frac{e^{-x}}{1+e^{-x}} \\ & =\frac{1}{1+e^{-x}} \cdot \frac{\left(1+e^{-x}\right)-1}{1+e^{-x}} \\ & =\frac{1}{1+e^{-x}} \cdot\left(\frac{1+e^{-x}}{1+e^{-x}}-\frac{1}{1+e^{-x}}\right) \\ & =\frac{1}{1+e^{-x}} \cdot\left(1-\frac{1}{1+e^{-x}}\right) \\ & =\sigma(x) \cdot(1-\sigma(x)) \end{aligned} $ --- ## References