Usefulness of extended domain of derivative function? Say I have $f(x)=\ln(x)$. We know its domain is $(0,\infty)$, but the domain of $f’(x)$ is $(-\infty,0) \cup (0,\infty)$. Though this is a relatively simple example, is there any application of the derivative extended beyond its function’s real domain? For example, does the derivative when $x$ is negative in this case have any implications in the complex plane, for some sort of analytic continuity, or is the derivative never used where the function is not defined on the reals?

Alex is right that the derivative has the same domain as the function, but what you have discovered is that the derivative may be easier to extend analytically than the original function. In the example of $\log$, one can extend the derivative $1/x$ to all complex numbers, which gives one a hint as to the correct generalization of the log function to the complex numbers as $\log(z) = \int_\gamma 1/z dz$, where $\gamma$ is an appropriately-chosen path from 1 to $z$.
In flavor, this is similar to applications in analytic number theory, where functions like the Riemann zeta function may have nice descriptions in terms of their derivatives on the real positive line, and this is a clue for how to proceed in the whole complex plane. I am not sure how precise this analogy turns out to be though.
