I'm sure Derek couldn't do it when he started playing piano for the first time. Improvisation grew out of the general ability to play piano for me, once I played various styles, memorized a lot of pieces, rhythms, building blocks of music, listening experience etc, improvisations could flow out without too much of a focused attention as to what notes to play and the direction the piece moves.
I think master improvisers develop this attribute, they can control what they do more so, but it might not actually be a necessary element of improvisation as a whole but rather a category of improvising. I love the freedom of improvisation like a meditation not trying to conform to a particular style or system, others like certain styles depending on their musical past and experience. Certainly it is very rewarding learning improvisation styles I love improv based on Cocktail lounge music, another rhythmic system I love is the Bosa Nova!
There is another side of improv where you let your minds eye hear sounds and the hands react. But this is a very fuzzy area to explain because the mind slips between the two when you attempt to train it. When you hear something in your minds eye then your hands play it, often the hands do not exactly create what your mind imagines so when your ears hear this it alters what the mind thinks about in the future! So the hands can effect the sound in your mind and the mind can effect the hand. This tug of war is very interesting but not something I have fully understood through my own improv exploration.