sigmoid

$$\sigma (z)=\frac{1}{1+e^{-z}}$$

Maps $\mathbb{R}\to (0,1)$, with $\sigma (0)=0.5$ and symmetry around that point. As a nonlinear function, it’s used in logistic regression to turn linear outputs into probabilities, and historically as an activation function in ANNs.

The derivative ${\sigma }^{\prime }(z)=\sigma (z)(1-\sigma (z))$ is convenient because you can compute it directly from the forward pass output $\sigma (z)$ without storing the input $z$ - which reduced memory overhead in early backpropagation implementations.