Logistic (Sigmoid)

\[ \sigma(x)=\frac{1}{1+\exp(-x)}\in(0,1), \] and $\sigma'(x)=\sigma(x)\left(1-\sigma(x)\right)$.

Tanh

\[ \tanh(x)=\frac{e^x-e^{-x}}{e^x+e^{-x}}\in(-1,1), \] and $\tanh'(x)=1-\tanh^2(x)$.

Tanh is just rescaled and shifted sigmoid, \[ \tanh(x)=2\sigma(2x)-1. \]