I distinctly remember the first time a lecturer used the "dx/dt" "symbol" in normal algebraic operations (that is, multiply both equation sides by dt and so on). I was so shocked it's actually not a elaborate differentation symbol, but something with actual division. Next time it was similar with integration, where the dx was substituted by some other function of du.

I swear I treated those as some grammar token, which doesn't hold any real meaning. I've been using those as such for years before.