What is the derivative of ln(x) and how does it relate to the graph of 1/x?

[NekotoKoara]

I feel like I might be missing something here, but how is the deriviative of ln(x) equal to 1/x? The graph of ln(x) only has valid x values to the right of the x axis. The graph for 1/x has valid values for x for all real numbers except for 0. Am I to deduce that functions and their derivatives do not always necessarily occupy the same domain? If so are the values to the left of the y-axis of any significance or should they just be ignored?

 

[fresh_42]

For $x < 0$ we get with the chain rule $\frac{d}{dx}\log (-x) = \frac{1}{-x}\cdot (-1) = \frac{1}{x}$ and both give $\frac{d}{dx}\log |x| = \frac{1}{x}$ for $x \neq 0$ which is the proper formula for the derivative of the logarithm function for both branches of $\frac{1}{x}$.