Doing derivatives without the definition is just calling on a black box.

A definition is a black box (that happens to have official status). The process I describe above leads, when managed with foresight, to an intuitive way to produce a definition. Sure, it may not include the slogan "brought to you by apostrophe, the letters LIM and an arrow" but you can go on to tell them "this is how impressive mathematcians say you should write this stuff that you already understand" and they'll get it.

I note that some people do learn best by having a black box definition shoved down their throats while others learn best by building from a solid foundation of understanding. Juggling both types isn't easy.