More seriously, it's about doing the impossible. Formally, some functions are nondifferentiable, period. But it would be cool if we could actually "more or less" differentiate them. For that we'll necessarily need a bag of tricks which is now coalescing into "techniques" and "principles".
Cf. numerical analysis. It takes a page or two to set up your definitions and show that many functions are badly conditioned, period. And yet we still want to compute them, so we've been building the bag of tricks for almost a century now.