I'm working through a derivative problem involving inverse trig functions from a calculus course on Udemy. The part of this problem I'm having trouble with is the simplification step at the very end. A snippet of the work-through is pictured below:
How is the expression with the red border simplified to the expression with the green border? In my mind, taking the common factor of x out of the absolute value expression changes the value of the surrounding expression for values of x < 0, right? I'm probably missing something... My math skills are rusty...
EDIT: Here's the original problem:
The expression on the left (in red) is nonnegative. The expression on the right (in green) is negative for negative values of x. Also, the answer is only real if $|x| > 1$. So they are only equal if x is assumed positive.