The sigmoid function is
f(x) =      1
         1 + e
when e is the exponential function and B is a parameter to adjust the steepness.

This function has the following properties:

One practical application of this function is the firing function in multi-layer perceptron neural networks.

It is used because of its simple derivative:

f'(x) =         -Bx
             B e
                -Bx  2
        ( 1 + e     )

      = B f(x) (1 - f(x))

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