Question

show that d/dx lxl = x/lxl

Answer 1

\[\text{ evaluate }x \ge 0; \frac{d}{dx}x=? \text{ and evaluate this } x<0 ; \frac{d}{dx}(-x)=? \]

Answer 2

would it have a derivative if it does not have a limit?

Answer 3

Use the piecewise definition for /x/

/x/ = x, where x is greater than or equal to 0
= -x, where x < 0

Answer 4

and that is x>0 sorry not x>=0 since |x| is not differentiable at x=0

Answer 5

after this step how can i make it = x/lxl then?

Answer 6

I'm pretty sure amistre demonstrated this a while back... use \[\Large |x| = \sqrt{x^2}\] and then differentiate using your chain rule et al..
.

Answer 7

or I did

Answer 8

and any calculus text book like Spivak's introduction to analysis would have