why do we have to rewrite as ?
Draw the graph of ln(x). It's derivative is always increasing so 1/x has to be positive for the domain of the original function. You can also draw the graph of 1/x. There is a positive and negative portion of the graph. The absolute value keeps only the positive portion. The first derivative tells you how the original function changes or how it's slope changes. As you get close to zero the slope gets really steep. Try putting really small positive numbers into 1/x. What happens?