I don't understand why the derivative of the natural [log] of 3x would equal 1/x...
Apply the Chain Rule. Check the steps and verify the cancellation.
...but the derivative of [the natural log of] 3x^(2)-1 is 1/3x^(2)-1
You might want to verify the actual derivative. The result displayed at your link does not match what you've posted.
To learn the "why", clearly write out each stage of the computation in each derivative. If you are unable to match the correct expressions, kindly please reply showing your work, so that errors may be located and explicated. Thank you.