Why isn't $\frac{\mathrm{d} }{\mathrm{d} x} \ln(x)$ specified as $\frac{1}{x},x>0$? As I understand, $\begin{eqnarray} \frac{\mathrm{d}}{\mathrm{d}x}\ln(x)\end{eqnarray} $ is generally specified as $\begin{eqnarray} \frac{1}{x} \end{eqnarray}$.  Would it be more appropriate to state it as $\begin{eqnarray} \frac{1}{x}, x>0\end{eqnarray}$ since $\ln(x)$ is undefined for $x\leq0$?  If not, why not?  In addition, what does this imply about the indefinite integral of or the definite integral with negative limits of integration of $\begin{eqnarray} \frac{1}{x} \end{eqnarray}$?
Thank you!

Another answer.  Some strange people use complex numbers, not just real numbers.  For them, log(x) makes sense other than for $x>0$.  And for them its derivative is $1/x$.  More importantly: for them, $\int (1/x)\,dx = \log(|x|)+C$ is WRONG!
