The Softmax function and its derivative

What happened

The softmax function takes an N-dimensional vector of arbitrary real values and produces another N-dimensional vector with real values in the range (0, 1) that add up to 1.0. It maps S(\mathbf{a}):\mathbb{R}^{N}\rightarrow \mathbb{R}^{N} :

Summary assembled by rule from the sources below

Sources