The way we teach calculus is based off of rote memorization. You need all the rules to solve the integrals. However, functional analysis is an almost entirely different kind of skill. Functional analysis is based on the theory that underlies calculus but that is usually skipped in order to just teach straight problem solving.
I see skills like functional analysis as more important since you learn what to expect from functions and why. The exact answer a computer can give you but a better understanding of functions will tell you very quickly if you made a major error in setting up the system on a computer, or if there are multiple answer how to determine which is the correct one for your system.
There is just not enough time to teach understanding (since it takes so much experience to gain it) and the memorization of rules for solving integrals and derivatives. Since any cellphone, laptop, tablet, etc can solve integrals and differentials but they can't give you understanding I think we should be spending time on the parts that computers can't do. As a result you can solve more realistic (and FAR harder) problems and you learn far more valuable skills in problem solving.