is |x|/x equal to x/|x|

Question

The derivative of absolute value is defined as x/|x| But of course that correspond to the sign of x. So then, why can't it be interchanged in exercises resolution, as also |x|/x is equivalent to the sign of x?

Answer 1

By definition of absolute value we have that

• for $x>0 \implies \frac{|x|}x=\frac x x=\frac x{|x|}=1$

• for $x<0 \implies \frac{|x|}x=-\frac x x=\frac x{|x|}=-1$

Answer 2

Note that for $x\ne 0,$ we have $$\frac{x}{|x|}=\frac{x}{|x|}\frac{|x|}{|x|}=\frac{x|x|}{|x|^2}=\frac{x|x|}{x^2}=\frac{|x|}{x}.$$