My argument is that it is worth memorising also the derivations, rather than re-deriving from scratch each time.

Meorising the derivation makes it easier to derive a second-order derivation, and so on. At some level of abstraction, going from first principles becomes prohibitively expensive and caching intermediary results, or so to speak, unlocks that again.

Sometimes yes, just like jargon is sometimes useful. Why use long-winded terms or descriptions when shorthand works between professionals.

Math proof and derivations are a bit like remembering a walking route. You've seen the start and end, and the main turns taken, and there's also a general "walking" skill you need.

This is similar to a loose life-thought i've had for a while, though i lack a catchy phrase for it lol.

My thought is: You're always practicing a behavior whether you like it or not.

Your mind is always setting you up to do more of whatever it is you're doing now. Both physical and mental. To do it more efficiently. With more ease. With more frequency. etc.

It's good motivation for me to mitigate a lot of negative behaviors. Angry in traffic, self anger, etc. If i can reason at least, of course. As i'm not interested in doing a lot of things more than i am, so i should avoid doing them now - if possible.