Question

When I differentiate a function like ln (1/x) I get something like (well exactly like) -1/x. But as with many derivatives ln functions, -1/x has two parts (it is a pair of hyperbolas) one of which does not, in its graph look like a plausible derivative of ln 1/x. Am I wrong if I fail to exclude that part when I give the answer -1/x? Or is there something subtle here I am missing?

Answer 1

The domain of the derivative of ln(1/x) would be equal to or a subset of the domain of ln(1/x). In other words, if we say -1/x is the derivative of ln(1/x) we can safely assume only the positive branch of the hyperbola is meant. But if you explicitly state the domain, that is helpful.