1. move the 3 in front of the x
2. where the 3 was, subtract 1 from it and put that back where the 3 was
but if you think of it as symbols you get more like
1. move the exponent symbol in front of the x symbol
2. subtract 1 from that exponent and put it back where the original exponent symbol was
For me at least, bridging the gap from the specifics of the first example to the more generalized second approach meant that when I was given something like
3x^(y^4) d/dx my mind was absolutely blown because I didn't know if there was some rule I needed to know if the exponent had an exponent or something.
Going from the years of the arithmetic approach to really groking the symbolic manipulation approach was really hard for me, and looking at lots of young kids suffering through algebra it was the same.
Learning how to recognize the symbols through the specificity of the numbers was something I don't think I really got at all until calculus and it's because the mechanics of the rules of simple derivatives and integration really force you to recognize the underlying symbols...more than algebra did (which for me just seems like a really complicated arithmetic rather than what's actually a pretty simple symbol manipulation exercise).
It's odd, because like most technically minded kids, I was already great working with symbols and breaking things down and building them up. I could build huge structures out of various small assemblies with legos, write simple software, crank out papers for English class that would guarantee an A and so on, but understanding that the teachers no longer wanted me to be a human tabulation device but to be a manipulator was something I never really cottoned on to.
I know from spending a little time with nieces and nephews younger then 10 that you can teach basic derivatives in about an hour or two and have them fairly reliably doing simple work like the above in an afternoon if you don't worry them too much about simplification of the result or all the arithmetic.
My gut says that if you start adding more rules, like what do you do when you have something like (x^3)/(x^2) and how exponents should be subtracted, and with a little care in the examples you show them of this rule in action, you end up showing them all of algebra, fractions, exponents and later trig, logs, etc. while building up and hanging all of these concepts off of calculus.
TBH I don't know enough about early childhood education to know if this should be used instead of basic arithmetic or not, but I bet if you use shapes and colors instead of letters and numbers and start to teach basic rules, you can just slowly introduce numeracy later anyways when their brains are more developed.