What is the most general antiderivative of $\displaystyle \frac{1}{x}$? I'll be you are expecting that:

$\displaystyle \int \frac{dx}{x} = ln|x| + C$.

BUT

Note that $\displaystyle \frac{d}{dx}ln ( -|x| ) = \frac{1}{x}$.

Now, I'll be the first to admit that $\displaystyle ln ( -|x| )$ has no real values. But itdoesmanage to solve the differential equation $\displaystyle y^{\prime} = \frac{1}{x}$.

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