Mathematically speaking, the object computes u and v partial derivatives of the norm of the output channel and assigns the computed two values to the output channel. Then it transforms the result to the surface orientation and normalizes the result. Practically speaking, these are exactly the steps that transform any function to a bump-map suitable form.
Usually, at least some of the objects use the same input channel (i.e. the channel with respect which the derivative is computed) as their parent 'Derive' object. Also, at least one of the sub objects should assign a value to the same output channel the derive object has (otherwise the derivative would be null).
The Derive object is mainly included for backward compatibility reasons. It has its use when computing a bump normal modification in the 'Surface properties' shader, but the same effect can be achieved in a simpler manner using a 'Surface geometry' shader.