UK A-levels certainly used to teach calculus quite successfully without ever deriving a single differential by hand. Students specializing in maths will have done differentiation using product and composition rule, integration, by parts and by substitution, along with trig identities. Basic classes of ODEs are also solved.

This is pretty good for the 80% who don't go on to study maths, and still a useful intro to the ones who redo it properly at Uni.