Oh Tospq, you are so affected by Physics, that you look for a meaning behind everything in the Formalistic Playground of Maths!
What is the most general antiderivative of ? I'll be you are expecting that:
Note that .
Now, I'll be the first to admit that has no real values. But it does manage to solve the differential equation .
So what, if anything, does this all mean? And what does it mean to have a function with no (real) domain, but that has a real valued derivative? There's something I'm just not picturing right here.
Another explanation is that you just cannot use the chain rule here. The "function" is simply now a function. It is like taking the derivative of .