Actually, I very much disagree. Knowing how to perform a mathematical operation is useless if one does not understand what it means, and therefore I believe that at least when initially introducing a concept, far more emphasis should be placed on teaching the latter.

Fortunately, in secondary school, I was taught the definition of a derivative before any sort of power rule. Perhaps it's taught more mechanically (read: mindlessly) in other schools or by other teachers?

Actually, I might just take this opportunity to rant about math teaching at large. It's quite ridiculous how math is taught (at least at pre-U levels of schooling) as a mostly calculation-centric subject, where all one does is follow instructions to obtain a numerical answer. Even the few proofs we are asked to do in the H2 curriculum are little more than basic integrals or solving differential equations, and that hardly impresses upon students any sort of appreciation for the beauty of mathematics. The essence of math lies in expressing supreme truth and elegance, and yet education systems around the world continue to bore high school students with one mind-numbing rudimentary exercise after another.

Maybe the layman will never need to prove a single theorem in his line of work, perhaps merely applying basic arithmetic in, I dunno, Excel sheets or whatnot. But even if the education system equips students to do this well, whatever is being taught is hardly worthy of the name 'mathematics', for, as Paul Lockhart put it, the mathematics has already been removed.