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:
.
BUT
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.
-Dan
The chain rule for works only for functions so that is differenciable and is differenciable on the range of . Now the range of over here is certainly is not differenciable on this interval.
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 .