Hello,

1. You've found three different possible relations between the domain of a function and the domain of the first derivative of this function. But the first case you mention is impossible:

If then there exist at least one x for which f(x) doesn't exist but the derivative. Or in plain words: That can't be.

The only relation between these two sets is:

2. Considering your examples:

i) because the limit exist: .

But: because ln(0) is not defined. So

ii) Obviously

iii) . The ln-function is defined for positive real numbers only and therefore the domain of the first derivation can't be greater than the domain of the function (see above). Thus